An example of a pair of angles that satisfies the given condition of "two obtuse adjacent angles" is Angle A and Angle B, where Angle A and Angle B are adjacent angles and both are obtuse.
Adjacent angles are two angles that share a common vertex and a common side but have no common interior points.
Obtuse angles are angles that measure greater than 90 degrees but less than 180 degrees.
To meet the given condition, we can consider Angle A and Angle B, where both angles are adjacent and both are obtuse.
Since the condition does not specify any specific measurements or orientations, we can assume any two adjacent obtuse angles to satisfy the condition.
For example, let Angle A be an obtuse angle measuring 110 degrees and Angle B be another obtuse angle measuring 120 degrees. These angles are adjacent as they share a common vertex and a common side, and both angles are obtuse since they measure more than 90 degrees.
Therefore, Angle A and Angle B form an example of a pair of "two obtuse adjacent angles" that satisfies the given condition.
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Monica’s number is shown below. In Monica’s number, how many times greater is the value of the 6 in the ten-thousands place than the value of the 6 in the tens place?
The value of the 6 in the ten-thousands place is 10,000 times greater than the value of the 6 in the tens place.
What is a place value?In Mathematics and Geometry, a place value is a numerical value (number) which denotes a digit based on its position in a given number and it includes the following:
TenthsHundredthsThousandthsUnitTensHundredsThousands.Ten thousands.6 in the ten-thousands = 60,000
6 in the tens place = 60
Value = 60,000/60
Value = 10,000.
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Consider the vectors u = (3,-4,-1) and v = (0,5,2). Find u v and determine the angle between u and v. [4] 1.2) Determine if the three vectors u = (1,4,-7), v = (2,-1, 4) and w = (0, -9, 18) lie in the same plane or not. [6] 1.3) Determine if the line that passes through the point (0, -3, -8) and is parallel to the line given by x = 10 + 3t, y = 12t and z=-3-t passes through the xz-plane. If it does give the coordinates of the point. [9] 1.4) Determine the equation of the plane that contains the points P = (1, -2,0), Q = (3, 1, 4) and Q = (0,-1,2) [8]
1.1)Consider the vectors u = (3,-4,-1) and v = (0,5,2). Find u v and determine the angle between u and v.
Solution:Given vectors areu = (3,-4,-1) and v = (0,5,2).The dot product of two vectors is given byu.v = |u||v|cosθ
where, θ is the angle between two vectors.Let's calculate u.vu.v = 3×0 + (-4)×5 + (-1)×2= -20
Hence, u.v = -20The magnitude of vector u is |u| = √(3² + (-4)² + (-1)²)= √26The magnitude of vector v is |v| = √(0² + 5² + 2²)= √29
Hence, the angle between u and v is given byu.v = |u||v|cosθcosθ = u.v / |u||v|cosθ = -20 / (√26 × √29)cosθ = -20 / 13∴ θ = cos⁻¹(-20 / 13)θ ≈ 129.8°The angle between vectors u and v is approximately 129.8°2.1)Determine if the three vectors u = (1,4,-7), v = (2,-1, 4) and w = (0, -9, 18) lie in the same plane or not.Solution:
To check whether vectors u, v and w lie in the same plane or not, we can check whether the triple scalar product is zero or not.The triple scalar product of vectors a, b and c is defined asa . (b × c)
Let's calculate the triple scalar product for vectors u, v and w.u . (v × w)u . (v × w) = (1,4,-7) . ((2, -1, 4) × (0,-9,18))u . (v × w) = (1,4,-7) . (126, 8, 18)u . (v × w) = 0Hence, u, v and w lie in the same plane.2.3)Determine if the line that passes through the point (0, -3, -8) and is parallel to the line given by x = 10 + 3t, y = 12t and z=-3-t passes through the xz-plane.
If it does give the coordinates of the point.Solution:We can see that the given line is parallel to the line (10,0,-3) + t(3,12,-1). This means that the direction ratios of both lines are proportional.
Let's calculate the direction ratios of the given line.The given line is parallel to the line (10,0,-3) + t(3,12,-1).Hence, the direction ratios of the given line are 3, 12, -1.We know that a line lies in a plane if the direction ratios of the line are proportional to the direction ratios of the plane.
Let's take the direction ratios of the xz-plane to be 0, k, 0.The direction ratios of the given line are 3, 12, -1. Let's equate the ratios to check whether they are proportional or not.3/0 = 12/k = -1/0We can see that 3/0 and -1/0 are not defined. But, 12/k = 12k/1Let's equate 12k/1 to 3/0.12k/1 = 3/0k = 0
Hence, the direction ratios of the given line are not proportional to the direction ratios of the xz-plane.
This means that the line does not pass through the xz-plane.2.4)Determine the equation of the plane that contains the points P = (1, -2,0), Q = (3, 1, 4) and Q = (0,-1,2).Solution:Let the required plane have the equationax + by + cz + d = 0Since the plane contains the point P = (1, -2,0),
substituting the coordinates of P into the equation of the plane givesa(1) + b(-2) + c(0) + d = 0a - 2b + d = 0This can be written asa - 2b = -d ---------------(1
)Similarly, using the points Q and R in the equation of the plane givesa(3) + b(1) + c(4) + d = 0 ---------------(2)and, a(0) + b(-1) + c(2) + d = 0 ---------------(3)E
quations (1), (2) and (3) can be written as the matrix equation shown below.[1 -2 0 0][3 1 4 0][0 -1 2 0][a b c d] = [0 0 0]
Let's apply row operations to the augmented matrix to solve for a, b, c and d.R2 - 3R1 → R2[-2 5 0 0][3 1 4 0][0 -1 2 0][a b c d] = [0 -3 0]R3 + R1 → R3[-2 5 0 0][3 1 4 0][0 3 2 0][a b c d] = [0 -3 0]3R2 + 5R1 → R1[-6 0 20 0][3 1 4 0][0 3 2 0][a b c d] = [-15 -3 0]R1/(-6) → R1[1 0 -3⅓ 0][3 1 4 0][0 3 2 0][a b c d] = [5/2 1/2 0]3R2 - R3 → R2[1 0 -3⅓ 0][3 -1 2 0][0 3 2 0][a b c d] = [5/2 -3/2 0]Now, let's solve for a, b, c and d.3b + 2c = 0[3 -1 2 0][a b c d] = [-3/2 1/2 0]a - (6/7)c = (5/14)[1 0 -3⅓ 0][a b c d] = [5/2 1/2 0]a + (3/7)c = (3/14)[1 0 -3⅓ 0][a b c d] = [1/2 1/2 0]a = 1/6(2) - 1/6(0) - 1/6(0)a = 1/3Hence,a = 1/3b = -2/3c = -1/7d = -5/7The equation of the plane that passes through the points P = (1, -2,0), Q = (3, 1, 4) and R = (0,-1,2) is given by1/3x - 2/3y - 1/7z - 5/7 = 0.
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what is the correct numerator for the derivative of after you have combined and and simplified the result but before you have factored an ‘h’ from the numerator.
The correct numerator for the derivative after we have combined and simplified the result but before we have factored an 'h' from the numerator is f(a+h)-f(a)-hf'(a).
In a given expression, if we combine and simplify the numerator of the derivative result but before we factor an 'h' from the numerator, then the correct numerator will be
f(a+h)-f(a)-hf'(a).
How do you find the derivative of a function? The derivative of a function can be calculated using various methods and notations such as using limits, differential, or derivatives using algebraic formulas.
Let's take a look at how to find the derivative of a function using the limit notation:
f'(a)=\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}
Here, f'(a) is the derivative of the function
f(x) at x=a.
To calculate the numerator of the derivative result, we can subtract
f(a) from f(a+h) to get the change in f(x) from a to a+h. This can be written as f(a+h)-f(a). Then we need to multiply the derivative of the function with the increment of the input, i.e., hf'(a).
Now, if we simplify and combine these two results, the correct numerator will be f(a+h)-f(a)-hf'(a)$. Therefore, the correct numerator for the derivative after we have combined and simplified the result but before we have factored an 'h' from the numerator is f(a+h)-f(a)-hf'(a).
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Can anyone help please
Answer:
The closest option from the given choices is option a) $84,000.
Step-by-step explanation:
Sales revenue: $100,000
Expenses: $10,000 (wages) + $3,000 (advertising) + $1,000 (dividends) + $3,000 (insurance) = $17,000
Profit = Sales revenue - Expenses
Profit = $100,000 - $17,000
Profit = $83,000
Therefore, the company made a profit of $83,000.
Find the foci for each equation of an ellipse.
16 x²+4 y²=64
For the equation 16x² + 4y² = 64, there are no real foci.
The foci for the equation of an ellipse, 16x² + 4y² = 64, can be found using the standard form equation of an ellipse. The equation represents an ellipse with its major axis along the x-axis.
To find the foci, we first need to determine the values of a and b, which represent the semi-major and semi-minor axes of the ellipse, respectively. Taking the square root of the denominators of x² and y², we have a = 2 and b = 4.
The formula to find the distance from the center to each focus is given by c = √(a² - b²). Substituting the values, we get c = √(4 - 16) = √(-12).
Since the square root of a negative number is imaginary, the ellipse does not have any real foci. Instead, the foci are imaginary points located along the imaginary axis. Therefore, for the equation 16x² + 4y² = 64, there are no real foci.
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Q3: Solve the given differential equation by using Variation of Parameters. x^2y" -2xy' + 2y = 1/x
The general solution to the given differential equation is:
y = y_c + y_p = C_1 + C_2x^3 + 1/x - 1/(8x^5)
We assume a solution of the form y_c = x^r. Plugging this into the homogeneous equation, we get:
r(r-1)x^r - 2rx^r + 2x^r = 0
r^2 - 3r = 0
This quadratic equation has two roots: r = 0 and r = 3. Therefore, the complementary solution is:
y_c = C_1x^0 + C_2x^3 = C_1 + C_2x^3
Next, we need to find the particular solution, which we assume as:
y_p = u_1(x)y_1(x) + u_2(x)y_2(x)
Here, y_1(x) = 1 and y_2(x) = x^3. To find u_1(x) and u_2(x),
formulas:
u_1(x) = -∫(y_2(x)f(x))/(W(x)) dx
u_2(x) = ∫(y_1(x)f(x))/(W(x)) dx
where f(x) = 1/x and W(x) is the Wronskian of y_1 and y_2.
Calculate:
u_1(x) = -∫(x^3/x)/(x^6) dx = -∫(1/x^2) dx = -(-1/x) = 1/x
u_2(x) = ∫(1/(x^3))/(x^6) dx = ∫(1/x^9) dx = -1/(8x^8)
Finally, the particular solution is given by:
y_p = (1/x)(1) + (-1/(8x^8))(x^3) = 1/x - 1/(8x^5)
Therefore, the general solution to the given differential equation is:
y = y_c + y_p = C_1 + C_2x^3 + 1/x - 1/(8x^5)
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Assume that there is a statistically significant bivariate relationship between the amount of texting during driving and the number of accidents. Scientists later investigate whether or not this bivariate relationship is moderated by age.
Age 16-20: r = 0.6 p = 0.01
Age 21+: r = 0.2 p = 0.05
T or F: Based only on the r and p values listed above you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.
It is False that based only on the r and p values listed above you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.
In the given scenario, it is not completely true that based only on the r and p values listed above, you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.
Let's first understand what is meant by the term "moderator.
"Moderator: A moderator variable is a variable that changes the strength of a connection between two variables. If there is a statistically significant bivariate relationship between the amount of texting during driving and the number of accidents, scientists investigate whether this bivariate relationship is moderated by age.
Therefore, based on the values of r and p, it is difficult to determine if age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.
As we have to analyze other factors also to determine whether the age is a moderator or not, such as the sample size, the effect size, and other aspects to draw a meaningful conclusion.
So, it is False that based only on the r and p values listed above you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.
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Use the method of variation of parameters to find a particular solution of the differential equation 4y" - 4y' + y = 80e¹/2 that does not involve any terms from the homogeneous solution. Y(t) = e. 40 t² ež. X
1. Homogeneous solution is [tex]\rm y_h(t) = c_1e^{(1/2t)} + c_2te^{(1/2t)[/tex].
2. Particular solution: [tex]\rm y_p(t) = 80e^{(1/2t)[/tex].
3. General solution: [tex]\rm y(t) = y_h(t) + y_p(t) = c_1e^{(1/2t)} + c_2te^{(1/2t)} + 80e^{(1/2t)[/tex].
1. Find the homogeneous solution:
The characteristic equation for the homogeneous equation is given by [tex]$4r^2 - 4r + 1 = 0$[/tex]. Solving this equation, we find that the roots are [tex]$r = \frac{1}{2}$[/tex] (double root).
Therefore, the homogeneous solution is [tex]$ \rm y_h(t) = c_1e^{\frac{1}{2}t} + c_2te^{\frac{1}{2}t}$[/tex], where [tex]$c_1$[/tex] and [tex]$c_2$[/tex] are constants.
2. Find the particular solution:
Assume the particular solution has the form [tex]$ \rm y_p(t) = u(t)e^{\frac{1}{2}t}$[/tex], where u(t) is a function to be determined. Differentiate [tex]$y_p(t)$[/tex] to find [tex]$y_p'$[/tex] and [tex]$y_p''$[/tex]:
[tex]$ \rm y_p' = u'e^{\frac{1}{2}t} + \frac{1}{2}ue^{\frac{1}{2}t}$[/tex]
[tex]$ \rm y_p'' = u''e^{\frac{1}{2}t} + u'e^{\frac{1}{2}t} + \frac{1}{4}ue^{\frac{1}{2}t}$[/tex]
Substitute these expressions into the differential equation [tex]$ \rm 4(y_p'') - 4(y_p') + y_p = 80e^{\frac{1}{2}}$[/tex]:
[tex]$ \rm 4(u''e^{\frac{1}{2}t} + u'e^{\frac{1}{2}t} + \frac{1}{4}ue^{\frac{1}{2}t}) - 4(u'e^{\frac{1}{2}t} + \frac{1}{2}ue^{\frac{1}{2}t}) + u(t)e^{\frac{1}{2}t} = 80e^{\frac{1}{2}}$[/tex]
Simplifying the equation:
[tex]$ \rm 4u''e^{\frac{1}{2}t} + u(t)e^{\frac{1}{2}t} = 80e^{\frac{1}{2}}$[/tex]
Divide through by [tex]$e^{\frac{1}{2}t}$[/tex]:
[tex]$4u'' + u = 80$[/tex]
3. Solve for u(t):
To solve for u(t), we assume a solution of the form u(t) = A, where A is a constant. Substitute this solution into the equation:
[tex]$4(0) + A = 80$[/tex]
[tex]$A = 80$[/tex]
Therefore, [tex]$u(t) = 80$[/tex].
4. Find the particular solution [tex]$y_p(t)$[/tex]:
Substitute [tex]$u(t) = 80$[/tex] back into [tex]$y_p(t) = u(t)e^{\frac{1}{2}t}$[/tex]:
[tex]$y_p(t) = 80e^{\frac{1}{2}t}$[/tex]
Therefore, a particular solution of the differential equation [tex]$4y'' - 4y' + y = 80e^{\frac{1}{2}}$[/tex] that does not involve any terms from the homogeneous solution is [tex]$y_p(t) = 80e^{\frac{1}{2}t}$[/tex].
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We consider the non-homogeneous problem y" - y = 4z-2 cos(x) +-2 First we consider the homogeneous problem y" - y = 0: 1) the auxiliary equation is ar² + br+c=r^2-r 2) The roots of the auxiliary equation are 3) A fundamental set of solutions is complementary solution y c1/1 + 02/2 for arbitrary constants c₁ and ₂. 0. (enter answers as a comma separated list). y= (enter answers as a comma separated list). Using these we obtain the the Next we seek a particular solution y, of the non-homogeneous problem y"-4-2 cos() +2 using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y/p= We then find the general solution as a sum of the complementary solution C13/1+ C2/2 and a particular solution: y=ye+Up. Finally you are asked to use the general solution to solve an IVP. 5) Given the initial conditions (0) 1 and y' (0) =-6 find the unique solution to the IVP
For the non-homogeneous problem y" - y = 4z - 2cos(x) +- 2, the auxiliary equation is ar² + br + c = r² - r.
The roots of the auxiliary equation are complex conjugates.
A fundamental set of solutions for the homogeneous problem is ye = C₁e^xcos(x) + C₂e^xsin(x).
Using these, we can find a particular solution using the method of undetermined coefficients.
The general solution is the sum of the complementary solution and the particular solution.
By applying the initial conditions y(0) = 1 and y'(0) = -6, we can find the unique solution to the initial value problem.
To solve the homogeneous problem y" - y = 0, we consider the auxiliary equation ar² + br + c = r² - r.
In this case, the coefficients a, b, and c are 1, -1, and 0, respectively. The roots of the auxiliary equation are complex conjugates.
Denoting them as α ± βi, where α and β are real numbers, a fundamental set of solutions for the homogeneous problem is ye = C₁e^xcos(x) + C₂e^xsin(x), where C₁ and C₂ are arbitrary constants.
Next, we need to find a particular solution to the non-homogeneous problem y" - y = 4z - 2cos(x) +- 2 using the method of undetermined coefficients.
We assume a particular solution of the form yp = Az + B + Ccos(x) + Dsin(x), where A, B, C, and D are coefficients to be determined.
By substituting yp into the differential equation, we solve for the coefficients A, B, C, and D. This gives us the particular solution yp.
The general solution to the non-homogeneous problem is y = ye + yp, where ye is the complementary solution and yp is the particular solution.
Finally, to solve the initial value problem (IVP) with the given initial conditions y(0) = 1 and y'(0) = -6, we substitute these values into the general solution and solve for the arbitrary constants C₁ and C₂.
This will give us the unique solution to the IVP.
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he Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year. brogram. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown: To ensure a balanced use of advertising media, radio advertisements must not exceed 40% of the total number of advertisernents authorited. In addition, television should account for at least 10% of the total number of advertisements authorized. (a) If the promotional budget is limited to $20,500, how many commercial messages should be run on each medium to maximize total audience contact? If your answer is zero enter " 0 ". What is the alocation of the budget among the three media? What is the total audience reached? What is the allocation of the budget among the three media? What is the total audience reached? (b) By how much would audience contact increase if an extra $100 were allocated to the promotional budget? Round your answer to the nearest whole number, Increase in audience coverage of approximately
a) The allocated budget for radio advertising is $8,200, for television advertising is $2,050, and for online advertising is $10,250. The maximum number of messages is 41 for radio, 4 for television, and 102 for online, reaching a total audience of 1,000,000.
b) If an extra $100 were allocated to the promotional budget, the audience contact would increase by approximately 1 message.
The first step in solving this problem is to determine the amount of money that can be allocated to each advertising medium based on the given budget.
To do this, we need to calculate the percentages for each medium. Since the budget is $20,500, we can allocate 40% of the budget to radio and 10% to television.
40% of $20,500 is $8,200, which can be allocated to radio advertising.
10% of $20,500 is $2,050, which can be allocated to television advertising.
The remaining amount, $20,500 - $8,200 - $2,050 = $10,250, can be allocated to online advertising.
Next, we need to determine the maximum number of commercial messages that can be run on each medium to maximize total audience contact.
Let's assume that the cost of running a commercial message on radio is $200, on television is $500, and online is $100.
To determine the maximum number of commercial messages, we divide the allocated budget for each medium by the cost of running a commercial message.
For radio: $8,200 (allocated budget) / $200 (cost per message) = 41 messages
For television: $2,050 (allocated budget) / $500 (cost per message) = 4 messages
For online: $10,250 (allocated budget) / $100 (cost per message) = 102.5 messages
Since we cannot have a fraction of a message, we need to round down the number of online messages to the nearest whole number. Therefore, the maximum number of online messages is 102.
The total audience reached can be calculated by multiplying the number of messages by the estimated audience for each medium.
For radio: 41 messages * 10,000 (estimated audience per message) = 410,000
For television: 4 messages * 20,000 (estimated audience per message) = 80,000
For online: 102 messages * 5,000 (estimated audience per message) = 510,000
The total audience reached is 410,000 + 80,000 + 510,000 = 1,000,000.
Now, let's move on to part (b) of the question. We need to determine how much the audience contact would increase if an extra $100 were allocated to the promotional budget.
To do this, we can calculate the increase in audience coverage for each medium by dividing the extra $100 by the cost per message.
For radio: $100 (extra budget) / $200 (cost per message) = 0.5 messages (rounded down to 0)
For television: $100 (extra budget) / $500 (cost per message) = 0.2 messages (rounded down to 0)
For online: $100 (extra budget) / $100 (cost per message) = 1 message
The total increase in audience coverage would be 0 + 0 + 1 = 1 message.
Therefore, if an extra $100 were allocated to the promotional budget, the audience contact would increase by approximately 1 message.
Please note that the specific numbers used in this example are for illustration purposes only and may not reflect the actual values in the original question.
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Assume that A is similar to an upper triangular matrix U, then det A is the product of all its eigenvalues (counting multiplicity). Please explain why.
If matrix A is similar to an upper triangular matrix U, then det A is the product of all its eigenvalues (counting multiplicity).
When two matrices are similar, it means they represent the same linear transformation under different bases. In this case, matrix A and upper triangular matrix U represent the same linear transformation, but U has a convenient triangular form.
The eigenvalues of a matrix represent the values λ for which the equation A - λI = 0 holds, where I is the identity matrix. These eigenvalues capture the characteristic behavior of the matrix in terms of its transformations.
For an upper triangular matrix U, the diagonal entries are its eigenvalues. This is because the determinant of a triangular matrix is simply the product of its diagonal elements. Each eigenvalue appears along the diagonal, and any other entries below the diagonal are necessarily zero.
Since A and U are similar matrices, they share the same eigenvalues. Thus, if U is upper triangular with eigenvalues λ₁, λ₂, ..., λₙ, then A also has eigenvalues λ₁, λ₂, ..., λₙ.
The determinant of a matrix is the product of its eigenvalues. Since A and U have the same eigenvalues, det A = det U = λ₁ * λ₂ * ... * λₙ.
Therefore, if A is similar to an upper triangular matrix U, the determinant of A is the product of all its eigenvalues, counting multiplicity.
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Question 12 of 17
Which of the following pairs of functions are inverses of each other?
A. f(x)=3(3)-10 and g(x)=+10
-8
B. f(x)= x=8+9 and g(x) = 4(x+8)-9
C. f(x) = 4(x-12)+2 and g(x)=x+12-2
4
OD. f(x)-3-4 and g(x) = 2(x+4)
3
Answer:
Step-by-step explanation:
To determine if two functions are inverses of each other, we need to check if their compositions result in the identity function.
Let's examine each pair of functions:
A. f(x) = 3(3) - 10 and g(x) = -8
To find the composition, we substitute g(x) into f(x):
f(g(x)) = 3(-8) - 10 = -34
Since f(g(x)) ≠ x, these functions are not inverses of each other.
B. f(x) = x + 8 + 9 and g(x) = 4(x + 8) - 9
To find the composition, we substitute g(x) into f(x):
f(g(x)) = 4(x + 8) - 9 + 8 + 9 = 4x + 32
Since f(g(x)) ≠ x, these functions are not inverses of each other.
C. f(x) = 4(x - 12) + 2 and g(x) = x + 12 - 2
To find the composition, we substitute g(x) into f(x):
f(g(x)) = 4((x + 12) - 2) + 2 = 4x + 44
Since f(g(x)) ≠ x, these functions are not inverses of each other.
D. f(x) = 3 - 4 and g(x) = 2(x + 4)
To find the composition, we substitute g(x) into f(x):
f(g(x)) = 3 - 4 = -1
Since f(g(x)) = x, these functions are inverses of each other.
Therefore, the pair of functions f(x) = 3 - 4 and g(x) = 2(x + 4) are inverses of each other.
15. Identify y− intercept for f(x)=2(x^2−5)+4. 16. Let f(x)=x^2 +10x+28−m, find m if the function only has 1 (ONE) x-intercept.
15. The y-intercept for the function f(x) = 2(x² - 5) + 4 is -6.
16. To have only one x-intercept, the value of m in the function f(x) = x² + 10x + 28 - m needs to be 3.
How to Find the Y-intercept of a Function?15. To find the y-intercept for the function f(x) = 2(x² - 5) + 4, we need to substitute x = 0 into the equation and solve for y.
Substituting x = 0 into the equation:
f(0) = 2(0² - 5) + 4
= 2(-5) + 4
= -10 + 4
= -6
Therefore, the y-intercept for the function f(x) = 2(x² - 5) + 4 is -6.
16. To find the value of m for which the function f(x) = x² + 10x + 28 - m has only one x-intercept, we need to consider the discriminant of the quadratic equation.
The discriminant is given by the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In this case, the quadratic equation is x² + 10x + 28 - m = 0, which implies a = 1, b = 10, and c = 28 - m.
For the quadratic equation to have only one x-intercept, the discriminant must be equal to zero (Δ = 0).
Setting Δ = 0 and substituting the values of a, b, and c:
(10)² - 4(1)(28 - m) = 0
100 - 4(28 - m) = 0
100 - 112 + 4m = 0
4m - 12 = 0
4m = 12
m = 3
Therefore, the value of m for which the function f(x) = x² + 10x + 28 - m has only one x-intercept is m = 3.
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15. y-intercept for the function f(x) = 2(x^2 - 5) + 4 is -6.
To find the y-intercept for the function f(x) = 2(x^2 - 5) + 4, we set x = 0 and solve for y.
Substituting x = 0 into the equation, we have:
f(0) = 2(0^2 - 5) + 4
= 2(-5) + 4
= -10 + 4
= -6
Therefore, the y-intercept for the function f(x) = 2(x^2 - 5) + 4 is -6.
16. function f(x) = x^2 + 10x + 28 - m has only one x-intercept, then the value of m should be 3.
To find the value of m if the function f(x) = x^2 + 10x + 28 - m has only one x-intercept, we need to consider the discriminant of the quadratic equation.
The discriminant (D) is given by D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
For the given equation f(x) = x^2 + 10x + 28 - m, we can see that a = 1, b = 10, and c = 28 - m.
To have only one x-intercept, the discriminant D should be equal to zero. Therefore, we have:
D = 10^2 - 4(1)(28 - m)
= 100 - 4(28 - m)
= 100 - 112 + 4m
= -12 + 4m
Setting D = 0, we have:
-12 + 4m = 0
4m = 12
m = 12/4
m = 3
Therefore, if the function f(x) = x^2 + 10x + 28 - m has only one x-intercept, then the value of m should be 3.
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AB and CD are parallel. What is m/7?
OA. 30°
OB. 110°
OC. 60°
OD. 130°
Step-by-step explanation:
Without a visual aid or more information about the diagram, it is difficult to determine the value of m/7. Please provide more details or information about the diagram.
Given the function P(1) - (16)(z + 4), find its y-intercept is its z-intercepts are 1 When z→→ [infinity], y> When I →→→ [infinity], y 0 Question Help: Video 0 -1 and I₂ = 6 xoo (Input + or for the answer) . x[infinity] (Input + or for the answer) with I₁I₂
The y-intercept of the function P(z) is -60.
To find the y-intercept of the function P(z), we need to evaluate P(0), which gives us the value of the function when z = 0.
For P(z) = (1 - 16)(z + 4), substituting z = 0:
P(0) = (1 - 16)(0 + 4) = (-15)(4) = -60
Therefore, the y-intercept of the function P(z) is -60.
The z-intercept is given as z₁ = 1, which means P(z₁) = P(1) = 0.
As for the behavior of the function as z approaches positive or negative infinity:
When z goes to positive infinity (z → +∞), the function P(z) approaches negative infinity (y → -∞).
When z goes to negative infinity (z → -∞), the function P(z) also approaches negative infinity (y → -∞).
The information provided about I₁ and I₂ is unclear, so I cannot provide specific answers regarding those variables. If you can provide additional information or clarify the question, I will be happy to assist you further.To find the y-intercept of the function P(z), we need to evaluate P(0), which gives us the value of the function when z = 0.
For P(z) = (1 - 16)(z + 4), substituting z = 0:
P(0) = (1 - 16)(0 + 4) = (-15)(4) = -60
The z-intercept is given as z₁ = 1, which means P(z₁) = P(1) = 0.
As for the behavior of the function as z approaches positive or negative infinity:
When z goes to positive infinity (z → +∞), the function P(z) approaches negative infinity (y → -∞).
When z goes to negative infinity (z → -∞), the function P(z) also approaches negative infinity (y → -∞).
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Quadrilateral ABCD is rotated 90 degrees clockwise about the origin. What are the coordinates of quadrilateral A'B'C'D?
Answer:
D
Step-by-step explanation:
(x,y)
so,it will change (-y,x)
A' (5,5) ,B'(5, 1) ,C'(2,1), D'(1,5).
PLEASE HELP !! Drop downs :
1: gets larger, gets smaller, stays the same
2: negative, positive
3: decreasing, increasing, constant
4: a horizontal asymptote, positive infinity, negative infinity
The appropriate options which fills the drop-down are as follows :
gets larger positive increasingpositive infinity Interpreting Exponential graphThe rate of change of the graph can be deduced from the shape and direction of the exponential line. As the interval values moves from left to right, the value of the slope given by the exponential line moves up, hence, gets bigger or larger.
The direction of the exponential line from left to right, means that the slope or rate of change is positive. Hence, the average rate of change is also positive.
Since we have a positive slope , we can infer that the graph's function would be increasing. Hence, the graph depicts an increasing function and will continue to approach positive infinity.
Hence, the missing options are : gets larger, positive, increasing and positive infinity.
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) Using convolution theorem, find 2s c-{To (s²+4)² (6 marks)
The convolution integral will give us the expression for c(t), (s² + 4)². To find the inverse Laplace transform of the function C(s) = (s² + 4)², we can utilize the convolution theorem.
According to the convolution theorem, the inverse Laplace transform of the product of two functions in the Laplace domain is equivalent to the convolution of their inverse Laplace transforms in the time domain.
Let's denote the inverse Laplace transform of (s² + 4)² as c(t).
We can rewrite the function C(s) as the product of two simpler functions: C(s) = (s² + 4) * (s² + 4).
Taking the inverse Laplace transform of both sides using the convolution theorem, we get: c(t) = (f * g)(t), where f(t) is the inverse Laplace transform of (s² + 4), and g(t) is the inverse Laplace transform of (s² + 4).
To find c(t), we need to determine the inverse Laplace transforms of (s² + 4) and (s² + 4). These can be obtained from Laplace transform tables or by applying standard techniques for inverse Laplace transforms.
Once we have the inverse Laplace transforms of f(t) and g(t), we can convolve them to find c(t) using the convolution integral:
c(t) = ∫[0 to t] f(t - τ) * g(τ) dτ.
Evaluating the convolution integral will give us the expression for c(t), which represents the inverse Laplace transform of (s² + 4)².
Please note that without specific values or additional information, it is not possible to provide an explicit expression for c(t) in this case.
The process described above outlines the general approach to finding the inverse Laplace transform using the convolution theorem.
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A construction worker needs to put a rectangular window in the side of a
building. He knows from measuring that the top and bottom of the window
have a width of 5 feet and the sides have a length of 12 feet. He also
measured one diagonal to be 13 feet. What is the length of the other
diagonal?
OA. 5 feet
OB. 13 feet
O C. 17 feet
OD. 12 feet
SUBMIT
The length of the other diagonal is 13 feet.
How to find the length of the other diagonalWe are given that:
Length of rectangular window = 12 feetWidth of rectangular window = 5 feetDiagonal length = 13 feetWe can also apply Pythagoras theorem to find the other length of the diagonal of a rectangle.
[tex]\rightarrow\text{c}^2=\text{a}^2+\text{b}^2[/tex]
[tex]\rightarrow13^2 = 12^2 + 5^2[/tex]
[tex]\rightarrow169= 144 + 25[/tex]
[tex]\rightarrow\sqrt{169}[/tex]
[tex]\rightarrow\bold{13 \ feet}[/tex]
Hence, the length of the other diagonal is 13 feet.
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Find an expression for a unit vector normal to the surface
x = 7 cos (0) sin (4), y = 5 sin (0) sin (4), z = cos (4)
for 0 in [0, 2л] and о in [0, л].
(Enter your solution in the vector form (*,*,*). Use symbolic notation and fractions where needed.)
27 cos(0) sin (4), sin(0) sin(4),2 cos(4)
n =
4 49 cos² (0) sin² (4) + 4 25 sin² (0) sin² (4) + 4 cos² (4
The unit vector normal to the surface is (√3/3, √3/3, √3/3)
a unit vector normal to the surface defined by the parametric equations x = 7cos(θ)sin(4), y = 5sin(θ)sin(4), and z = cos(4), we need to calculate the gradient vector of the surface and then normalize it to obtain a unit vector.
The gradient vector of a surface is given by (∂f/∂x, ∂f/∂y, ∂f/∂z), where f(x, y, z) is an implicit equation of the surface. In this case, we can consider the equation f(x, y, z) = x - 7cos(θ)sin(4) + y - 5sin(θ)sin(4) + z - cos(4) = 0, as it represents the equation of the surface.
Taking the partial derivatives, we have:
∂f/∂x = 1
∂f/∂y = 1
∂f/∂z = 1
Therefore, the gradient vector is (1, 1, 1).
To obtain a unit vector, we need to normalize the gradient vector. The magnitude of the gradient vector is given by:
|∇f| = √(1^2 + 1^2 + 1^2) = √3.
Dividing the gradient vector by its magnitude, we have:
n = (1/√3, 1/√3, 1/√3).
Simplifying the expression, we get:
n = (√3/3, √3/3, √3/3).
Therefore, the unit vector normal to the surface is (√3/3, √3/3, √3/3).
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You need to provide a clear and detailed solution for the following questions: Question 1 : a) : Verify that the differential equation is exact: (-y sin(x)+7x6y³)dx+(8y7 cos(x)+3x7y²)dy = 0. b) : Find the general solution to the above differential equation. Question 2 : a) : Solve the following linear system in detailed, by using Gauss-Jordan elimination: x-3y - 5z = 2 2x + 5y-z = 1 x + 3y - 3z = -5 b) Is the system homogeneous and consistent? What about the solution type? Is it unique ? Question 3 : Let -3x - 6y=k² + 3k - 18 -6x - 3v = k²-9k +18 Question 3 : Let -3x - 6y = k² + 3k - 18 -6x - 3y = k² - 9k + 18 be a system of equations. a) : If the system is homogeneous, what is the value(s) for k ? b) : Solve the homogeneous system. Is the solution trivial? Is the solution unique ?
1a: The given differential equation is not exact.
1b: The general solution to the above differential equation is y = (x^7 - C)/(7x^6), where C is an arbitrary constant.
2a: The solution to the linear system using Gauss-Jordan elimination is x = 1, y = -1, z = -1.
2b: The system is homogeneous and consistent. The solution is unique.
For Question 1a, to determine if a differential equation is exact, we need to check if the partial derivatives of the coefficients with respect to the variables satisfy a certain condition. In this case, the equation is not exact because the partial derivative of (-y sin(x)+7x^6y³) with respect to y is not equal to the partial derivative of (8y^7 cos(x)+3x^7y²) with respect to x.
Moving on to Question 1b, we can find the general solution by integrating the equation. Integrating the terms with respect to their respective variables, we obtain y = (x^7 - C)/(7x^6), where C is the constant of integration. This represents the family of solutions to the given differential equation.
In Question 2a, we are asked to solve a linear system using Gauss-Jordan elimination. By performing the necessary row operations, we find the solution x = 1, y = -1, and z = -1.
Regarding Question 2b, the system is homogeneous because the right-hand side of each equation is zero. The system is consistent because it has a solution. Furthermore, the solution is unique since there are no free variables in the system after performing Gauss-Jordan elimination.
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Lush Gardens Co. bought a new truck for $56,000. It paid $5,600 of this amount as a down payment and financed the balance at 5.50% compounded semi-annually. If the company makes payments of $1,800 at the end of every month, how long will it take to settle the loan? years months Express the answer in years and months, rounded to the next payment period
It will take Lush Gardens Co. approximately 37 months to settle the loan.
To determine how long it will take for Lush Gardens Co. to settle the loan, we can use the formula for the future value of an ordinary annuity:
FV = P. ((1+r)ⁿ - 1)/r
Where:
FV is the future value of the annuity (the remaining loan balance)
P is the monthly payment
r is the interest rate per compounding period
n is the number of compounding periods
In this case, Lush Gardens Co. made a down payment of $5,600, leaving a balance of $56,000 - $5,600 = $50,400 to be financed.
The monthly payment (P) is $1,800.
The interest rate (r) is 5.50% per year, compounded semi-annually. To convert it to a monthly interest rate, we divide it by 12:
r = 5.50/100.12 = 0.004583
Let's calculate the number of compounding periods (n) required to settle the loan:
n = log(FV.r/p + 1)/log(r+1)
Substituting the given values into the equation, we can solve for n:
n = log(50,400×0.004583/1800 + 1)/log(0.004583+1)
we find that n is approximately 36.77 compounding periods. Since we make payments at the end of every month, we can round up to the next payment period.
Therefore, it will take Lush Gardens Co. approximately 37 months to settle the loan.
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Please do C and D. Thanks so much 2. (Exercise with summation)
In this exercise you will prove that the pattern of numbers on the right below, an, is equal to n³. Two potential solutions have been outlined for you below. Pick one.
= a1 a2 3+5 7+9+11 13+ 15 +17+ 19 = = = a4
21+23+25+27 +29 = a5 student submitted image, transcription available below
This path is more succint, but demands very precise language.
(a) Find an explicit formula R(n) for the rightmost odd number on the left hand side of the nth row above. For example, R(2) should yield 5, R(3) should be 11, and so on. Justify this formula - you must be able to prove this works always, not just for the first few.
(b) Now find a formula L(n) for the left most odd number in the nth row above. (So L(2) = 3, L(3) = 7). Justify this formula as well.
(c) How many odd numbers are on the left hand side in the nth row above?
(d) Using the previous three steps and the fact that each row has an even distribution to make an argument for what the value of an should be. This needs to be formally justified
(a) The explicit formula R(n) = 2n - 1.
(b) L(n) = n(n - 1).
(c) Number of odd numbers = 1 - n² + 3n.
(d) an = n³ + 2n² + n + 2.
(a) The explicit formula R(n) for the rightmost odd number on the left-hand side of the nth row, let's examine the pattern. In each row, the number of odd numbers on the left side is equal to the row number (n).
The first row (n = 1) has 1 odd number: a1.
The second row (n = 2) has 2 odd numbers: a2 and 3.
The third row (n = 3) has 3 odd numbers: 5, 7, and 9.
We can observe that in the nth row, the first odd number is given by n, and the subsequent odd numbers are consecutive odd integers. Therefore, we can express R(n) as:
R(n) = n + (n - 1) = 2n - 1.
To justify this formula, we can use mathematical induction. First, we verify that R(1) = 1, which matches the first row. Then, assuming the formula holds for some arbitrary kth row, we can show that it holds for the (k+1)th row:
R(k+1) = k + 1 + k = 2k + 1.
Since 2k + 1 is the (k+1)th odd number, the formula holds for the (k+1)th row.
(b) The formula L(n) for the leftmost odd number in the nth row, we can observe that the leftmost odd number in each row is given by the sum of odd numbers from 1 to (n-1). We can express L(n) as:
L(n) = 1 + 3 + 5 + ... + (2n - 3).
To justify this formula, we can use the formula for the sum of an arithmetic series:
S = (n/2)(first term + last term).
In this case, the first term is 1, and the last term is (2n - 3). Plugging these values into the formula, we have:
S = (n/2)(1 + 2n - 3) = (n/2)(2n - 2) = n(n - 1).
Therefore, L(n) = n(n - 1).
(c) The number of odd numbers on the left-hand side in the nth row can be calculated by subtracting the leftmost odd number from the rightmost odd number and adding 1. Therefore, the number of odd numbers in the nth row is:
Number of odd numbers = R(n) - L(n) + 1 = (2n - 1) - (n(n - 1)) + 1 = 2n - n² + n + 1 = 1 - n² + 3n.
(d) Based on the previous steps and the fact that each row has an even distribution of odd numbers, we can argue that the value of an, which represents the sum of odd numbers in the nth row, should be equal to the sum of the odd numbers in that row. Using the formula for the sum of an arithmetic series, we can find the sum of the odd numbers in the nth row:
Sum of odd numbers = (Number of odd numbers / 2) * (First odd number + Last odd number).
Sum of odd numbers = ((1 - n² + 3n) / 2) * (L(n) + R(n)).
Substituting the formulas for L(n) and R(n) from earlier, we get:
Sum of odd numbers = ((1 - n² + 3n) / 2) * (n(n - 1) + 2
n - 1).
Simplifying further:
Sum of odd numbers = (1 - n² + 3n) * (n² - n + 1).
Sum of odd numbers = n³ - n² + n - n² + n - 1 + 3n² - 3n + 3.
Sum of odd numbers = n³ + 2n² + n + 2.
Hence, the value of an is given by the sum of the odd numbers in the nth row, which is n³ + 2n² + n + 2.
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A firm issues three-month commercial paper with a $1000000
face value and pays an EAR of 7.4%. What is the amount the firm
receives?
If firm issues commercial paper with $1000000 face-value and pays EAR of 7.4%, then amount the firm will receive is $981500.
To calculate the amount the firm receives from issuing the three-month commercial paper, we need to determine the total interest earned over the three-month period.
The Effective Annual Rate (EAR) of 7.4% indicates the annualized interest rate. Since the commercial paper has 3-month term, we adjust the EAR to account for the shorter period.
To find the quarterly interest rate, we divide the EAR by the number of compounding periods in a year. In this case, since it is a 3-month period, there are 4-compounding periods in a year (quarterly compounding).
Quarterly interest rate = (EAR)/(number of compounding periods)
= 7.4%/4
= 1.85%,
Now, we calculate interest earned on "face-value" of $1,000,000 over 3-months,
Interest earned = (face value) × (quarterly interest rate)
= $1,000,000 × 1.85% = $18,500,
So, amount firm receives from issuing 3-month commercial paper is the face value minus the interest earned:
Amount received = (face value) - (interest earned)
= $1,000,000 - $18,500
= $981,500.
Therefore, the amount that firms receives is $981500.
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The heights of 10 women, in \( \mathrm{cm} \), are \( 168,160,168,154,158,152,152,150,152,150 \). Determine the mean. A. 153 B. 155 C. 152 D. \( 156.4 \)
The mean height of 10 women to the nearest whole number is 156.
In statistics, the mean is a measure of central tendency that represents the average value of a set of data points. It is calculated by summing up all the values in the dataset and dividing the sum by the total number of data points.
To determine the mean (average) height of the 10 women, you need to sum up all the heights and divide the total by the number of women. Let's calculate it:
Sum of heights = 168 + 160 + 168 + 154 + 158 + 152 + 152 + 150 + 152 + 150 = 1556
Number of women = 10
Mean height = Sum of heights / Number of women = 1556 / 10 = 155.6
Rounding the mean height to the nearest whole number, we get 156.
Therefore, the correct answer is D. 156.
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Perform the indicated operation and simplify: (26x+5)−(−4x2−13x+5) A) 4x2−39x B) 4x2+39x C) 4x2+39x−10 D) 4x2+13x+10 E) −4x2+13x+10
The solution for this question is [tex]A) 4�2−39�4x 2 −39x.[/tex]
To perform the indicated operation and simplify [tex]\((26x+5) - (-4x^2 - 13x + 5)\),[/tex]we distribute the negative sign to each term within the parentheses:
[tex]\((26x + 5) + 4x^2 + 13x - 5\)[/tex]
Now we can combine like terms:
[tex]\(26x + 5 + 4x^2 + 13x - 5\)[/tex]
Combine the[tex]\(x\)[/tex] terms: [tex]\(26x + 13x = 39x\)[/tex]
Combine the constant terms: [tex]\(5 - 5 = 0\)[/tex]
The simplified expression is [tex]\(4x^2 + 39x + 0\),[/tex] which can be further simplified to just [tex]\(4x^2 + 39x\).[/tex]
Therefore, the correct answer is A) [tex]\(4x^2 - 39x\).[/tex]
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rewrite the expression with a rational exponent as a radical expression. (1 point) five to the three fourths power all raised to the two thirds power
The expression "five to the three-fourths power raised to the two-thirds power" can be rewritten as a radical expression.
First, let's calculate the exponentiation inside the parentheses:
(5^(3/4))^2/3
To simplify this, we can use the property of exponentiation that states raising a power to another power involves multiplying the exponents:
5^((3/4) * (2/3))
When multiplying fractions, we multiply the numerators and denominators separately:
5^((3 * 2)/(4 * 3))
Simplifying further:
5^(6/12)
The numerator and denominator of the exponent can be divided by 6, which results in:
5^(1/2)
Now, let's express this in radical form. Since the exponent 1/2 represents the square root, we can write it as:
√5
Therefore, the expression "five to the three-fourths power raised to the two-thirds power" simplifies to the radical expression √5.
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1. Write the negation for each of the following statements a. All tests came back positive. b. Some tests came back positive. c. Some tests did not come back positive. d. No tests came back positive.
The negations for each of the following statements are as follows:
a. None of the tests came back positive.
b. No tests came back positive.
c. All tests came back positive.
d. Some tests came back positive.
Statement a. All tests came back positive.The negation of the statement is: None of the tests came back positive.
Statement b. Some tests came back positive.The negation of the statement is: No tests came back positive.
Statement c. Some tests did not come back positive.The negation of the statement is: All tests came back positive.
Statement d. No tests came back positive.The negation of the statement is: Some tests came back positive.
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2 3 4 6. Given matrix A = 4 3 1 1 2 4 (a) Calculate the determinant of A.
(b) Calculate the inverse of A by using the formula involving the adjoint of A.
(a) The determinant of matrix A is 5.
(b) The inverse of matrix A using the adjoint formula is [2/5 -3/5; -1/5 4/5].
How to calculate the determinant of matrix A?(a) To calculate the determinant of matrix A, denoted as |A| or det(A), we can use the formula for a 2x2 matrix:
det(A) = (a*d) - (b*c)
For matrix A = [4 3; 1 2], we have:
det(A) = (4*2) - (3*1)
= 8 - 3
= 5
Therefore, the determinant of matrix A is 5.
How to calculate the inverse of matrix A using the formula involving the adjoint of A?(b) To calculate the inverse of matrix A using the formula involving the adjoint of A, we follow these steps:
Calculate the determinant of A, which we found to be 5.
Find the adjoint of A, denoted as adj(A), by swapping the elements along the main diagonal and changing the sign of the off-diagonal elements. For matrix A, the adjoint is:
adj(A) = [2 -3; -1 4]
Calculate the inverse of A, denoted as A^(-1), using the formula:
[tex]A^{(-1)}[/tex] = (1/det(A)) * adj(A)
Plugging in the values, we have:
[tex]A^{(-1)}[/tex] = (1/5) * [2 -3; -1 4]
= [2/5 -3/5; -1/5 4/5]
Therefore, the inverse of matrix A is:
[tex]A^{(-1)}[/tex]= [2/5 -3/5; -1/5 4/5]
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Help!!!!!!!!!!!!!!!!!
Answer:
A. 6,000 units²
Step-by-step explanation:
A = LW
A = 100 units × 60 units
A = 6000 units²