The step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 0.5.
To compute the step length 'a' using the line search method, we can follow these steps:
1: Calculate the gradient at point x(k).
- Given x(k) = (1, -1)
- Compute the gradient ∇f(x₁,x₂) at x(k):
∇f(x₁,x₂) = (∂f/∂x₁, ∂f/∂x₂)
∂f/∂x₁ = 3x₁ + 1
∂f/∂x₂ = 2x₂ - 1
Substituting x(k) = (1, -1):
∂f/∂x₁ = 3(1) + 1 = 4
∂f/∂x₂ = 2(-1) - 1 = -3
- Gradient at x(k): ∇f(x(k)) = (4, -3)
2: Compute the dot product between the gradient and the descent direction.
- Given PL = (1, 0)
- Dot product: ∇f(x(k)) ⋅ PL = (4)(1) + (-3)(0) = 4
3: Compute the norm of the descent direction.
- Norm of PL: ||PL|| = √(1² + 0²) = √1 = 1
4: Calculate the step length 'a'.
- Step length formula: a = -∇f(x(k)) ⋅ PL / ||PL||²
a = -4 / (1²) = -4 / 1 = -4
5: Take the absolute value of 'a' to ensure a positive step length.
- Absolute value: |a| = |-4| = 4
6: Finalize the step length.
- The step length 'a' is the positive value of |-4|, which is 4.
Therefore, the step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 4.
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What is -2.93(b + 12) = -11.72
What is b
(Solve two-step linear equations)
How would you describe the difference between the graphs of f (x) = 3x²
and g(x) = -2² ?
OA. g(x) is a reflection of f(x) over the line y = x.
B. g(x) is a reflection of f(x) over the line y = -1.
C. g(x) is a reflection of f(x) over the x-axis.
D. g(x) is a reflection of f(x) over the y-axis.
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C).
The given functions are f(x) = 3x² and g(x) = -2².
To understand the difference between their graphs, let's examine the characteristics of each function individually:
Function f(x) = 3x²:
The coefficient of x² is positive (3), indicating an upward-opening parabola.
The graph of f(x) will be symmetric with respect to the y-axis, as any change in x will result in the same y-value due to the squaring of x.
The vertex of the parabola will be at the origin (0, 0) since there are no additional terms affecting the position of the graph.
Function g(x) = -2²:
The coefficient of x² is negative (-2), indicating a downward-opening parabola.
The negative sign will reflect the graph of f(x) across the x-axis, resulting in a vertical flip.
The vertex of the parabola will also be at the origin (0, 0) due to the absence of additional terms.
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C). This means that g(x) is obtained by taking the graph of f(x) and flipping it vertically. The reflection occurs over the x-axis, causing the parabola to open downward instead of upward.
Therefore, the correct answer is option C: g(x) is a reflection of f(x) over the x-axis.
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please help!!!!!!!!!!!!!!!!!!!!!!
The systematic sample would be A. The city manager takes a list of the residents and selects every 6th resident until 54 residents are selected.
The random sample would be C. The botanist assigns each plant a different number. Using a random number table, he draws 80 of those numbers at random. Then, he selects the plants assigned to the drawn numbers. Every set of 80 plants is equally likely to be drawn using the random number table.
The cluster sample is C. The host forms groups of 13 passengers based on the passengers' ages. Then, he randomly chooses 6 groups and selects all of the passengers in these groups.
What are systematic, random and cluster samples ?A systematic sample involves selecting items from a larger population at uniform intervals. A random sample involves selecting items such that every individual item has an equal chance of being chosen.
A cluster sample involves dividing the population into distinct groups (clusters), then selecting entire clusters for inclusion in the sample.
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 Ex is tangent to circle O at point L, and IF is a secant line. If m_FLX = 104°, find
mLKF.
Answer:
arc LKF = 208°
Step-by-step explanation:
the angle FLX between the tangent and the secant is half the measure of the intercepted arc LKF , then intercepted arc is twice angle FLX , so
arc LKF = 2 × 104° = 208°
Solve the system of equations using elimination.
5x + 3y = 8
4x + y = 12
O (1, 1)
O (2.4)
O (3,0)
O (4,-4)
Answer: O (4, -4)
Step-by-step explanation:
To solve the system of equations using elimination, we can multiply the second equation by -3 to eliminate the y term:
Original equations:
5x + 3y = 8 (Equation 1)
4x + y = 12 (Equation 2)
Multiply Equation 2 by -3:
-3(4x + y) = -3(12)
-12x - 3y = -36 (Equation 3)
Now we can add Equation 1 and Equation 3 to eliminate the y term:
(5x + 3y) + (-12x - 3y) = 8 + (-36)
Simplifying:
5x - 12x + 3y - 3y = 8 - 36
-7x = -28
Divide both sides by -7:
x = -28 / -7
x = 4
Now substitute the value of x back into either of the original equations, let's use Equation 2:
4(4) + y = 12
16 + y = 12
y = 12 - 16
y = -4
Therefore, the solution to the system of equations is x = 4 and y = -4.
Alonso brings
$
21
$21dollar sign, 21 to the market to buy eggs and avocados. He gets eggs that cost
$
2.50
$2.50dollar sign, 2, point, 50. Then, he notices that the store only sells avocados in bags of
3
33 for
$
5
$5dollar sign, 5. He wants to buy as many avocados as he can with his remaining money.
Let
�
BB represent the number of bags of avocados that Alonso buys.
Alonso spent all of his money, this confirms that he can buy 3 bags of avocados.
Alonso has $21.00 to spend on eggs and avocados. He buys eggs that cost $2.50, which leaves him with $18.50. Since the store only sells avocados in bags of 3, he will need to find the cost per bag in order to calculate how many bags he can buy.
First, divide the cost of 3 avocados by 3 to find the cost per avocado. $5.00 ÷ 3 = $1.67 per avocado.
Next, divide the money Alonso has left by the cost per avocado to find how many avocados he can buy.
$18.50 ÷ $1.67 per avocado = 11.08 avocados.
Since avocados only come in bags of 3, Alonso needs to round down to the nearest whole bag. He can buy 11 avocados, which is 3.67 bags.
Thus, he will buy 3 bags of avocados.Let's test our answer to make sure that Alonso has spent all his money:
$2.50 for eggs3 bags of avocados for $5.00 per bag, which is 9 bags of avocados altogether. 9 bags × $5.00 per bag = $45.00 spent on avocados.
Total spent:
$2.50 + $45.00 = $47.50
Total money had:
$21.00
Remaining money:
$0.00
Since Alonso spent all of his money, this confirms that he can buy 3 bags of avocados.
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Circumference of circle inscribed or circumscribed polygon
Hint: you will need to find the diameter of the circle, use Pythagorean Theorem)
ind then I out of the 3 problems.
Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.
8 cm
15 cm
The exact circumferences of the inscribed and circumscribed circles for the given polygons are 8π cm and 15π cm, respectively.
To find the exact circumference of a circle inscribed or circumscribed by a polygon, we can use the Pythagorean theorem to determine the diameter of the circle.
In the case of an inscribed polygon, the diameter of the circle is equal to the diagonal of the polygon. Let's consider the polygon with a diagonal of 8 cm. If we draw a line connecting two non-adjacent vertices of the polygon, we get a diagonal that represents the diameter of the inscribed circle.
Using the Pythagorean theorem, we can find the length of this diagonal. Let's assume the sides of the polygon are a and b. Then the diagonal can be found using the equation: diagonal^2 = a^2 + b^2. Substituting the given values, we have 8^2 = a^2 + b^2. Solving this equation, we find that a^2 + b^2 = 64.
For the circumscribed polygon with a diagonal of 15 cm, the diameter of the circle is equal to the longest side of the polygon. Let's assume the longest side of the polygon is c. Therefore, the diameter of the circumscribed circle is 15 cm.
Once we have determined the diameter of the circle, we can calculate its circumference using the formula C = πd, where C is the circumference and d is the diameter.
For the inscribed circle, the circumference would be C = π(8) = 8π cm.
For the circumscribed circle, the circumference would be C = π(15) = 15π cm.
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The product of 3, and a number increased by -7, is -36
┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈
✦ The number is - 5
┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈
[tex]\begin{gathered} \; \sf{\color{pink}{Let \; the \; other \; number \; be \; (x)::}} \\ \end{gathered}[/tex]
Atq,,
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x + ( - 7) \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x - 7 \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x - 21 = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 36 + 21} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 15} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{x = \dfrac{\cancel{ - 15}}{\cancel{ \: 3}}} \qquad \bigg \lgroup \sf{Cancelling \: by \: 3} \bigg \rgroup \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{pink} :\dashrightarrow \underline{\color{pink}\boxed{\colorbox{black}{x = - 5}}} \: \pmb{\bigstar} \\ \\ \end{gathered}[/tex]
The answer is:
z = -5Work/explanation:
The product means we multiply two numbers.
Here, we multiply 3 and a number increased by -7; let that number be z.
So we have
[tex]\sf{3(z+(-7)}[/tex]
simplify:
[tex]\sf{3(z-7)}[/tex]
This equals -36
[tex]\sf{3(z-7)=-36}[/tex]
[tex]\hspace{300}\above2[/tex]
[tex]\frak{solving~for~z}[/tex]
Distribute
[tex]\sf{3z-21=-36}[/tex]
Add 21 on each side
[tex]\sf{3z=-36+21}[/tex]
[tex]\sf{3z=-15}[/tex]
Divide each side by 3
[tex]\boxed{\boxed{\sf{z=-5}}}[/tex]
Darla, Ellie, and Fran ate a whole container of ice cream. Darla ate half as much as Ellie ate, and Fran ate 5 times as much as Darla ate. If the container of ice cream cost $4.00, how much, in dollars, should each person pay?
An isosceles triangle has two angles both equal to x. The third angle is 45 degrees bigger than either of these. Find the value of x.
Let's use the fact that the sum of the angles of a triangle is always 180 degrees to solve this problem. Let the two equal angles be x, then the third angle is x + 45.Let's add all the angles together:x + x + x + 45 = 180Simplifying this equation, we get:3x + 45 = 180Now, we need to isolate the variable on one side of the equation. We can do this by subtracting 45 from both sides of the equation:3x = 135Finally, we can solve for x by dividing both sides of the equation by 3:x = 45Therefore, the value of x is 45 degrees.
Answer:
45°
Step-by-step explanation:
An isosceles triangle has two angles both equal to x. The third angle is 45 degrees bigger than either of these. Find the value of x.Let's turn the question into an equation
180 = x + x + x + 45
180 - 45 = 3x
135 = 3x
x = 135 : 3
x = 45°
------------------
check
180 = 45 + 45 + 45 + 45
180 = 180
same value the answer is good
GEOMETRY 100POINTSSS
Find x
Answer:
5.9
Step-by-step explanation:
sin Θ = opp/hyp
sin 36° = x/10
x = 10 × sin 36°
x = 5.88
Answer: 5.9
93-(15x10)+(160:16) =
Answer:
Step-by-step explanation:
Let's calculate the expression step by step:
93 - (15 × 10) + (160 ÷ 16)
First, we perform the multiplication:
93 - 150 + (160 ÷ 16)
Next, we perform the division:
93 - 150 + 10
Finally, we perform the subtraction and addition:
-57 + 10
The result is:
-47
Therefore, 93 - (15 × 10) + (160 ÷ 16) equals -47.
Determine the equation of the ellipse with foci... 100points
The equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is[tex](x^2/289) + (y^2/225) = 1.[/tex]
To determine the equation of an ellipse given its foci and the length of its major axis, we need to use the standard form equation for an ellipse. The standard form equation for an ellipse centered at the origin is:
[tex](x^2/a^2) + (y^2/b^2) = 1[/tex]
where 'a' represents the semi-major axis and 'b' represents the semi-minor axis.
In this case, we know that the distance between the foci is equal to 2a, which means a = 34/2 = 17. The foci of the ellipse are given as (7, 17) and (7, -13). The foci lie on the major axis of the ellipse, and since their y-coordinates differ by 30 (17 - (-13) = 30), the length of the major axis is equal to 2b, which means b = 30/2 = 15.
Now we have the values of a and b, so we can substitute them into the standard form equation:
[tex](x^2/17^2) + (y^2/15^2) = 1[/tex]
Simplifying further, we have:
[tex](x^2/289) + (y^2/225) = 1[/tex]
Therefore, the equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is:
[tex](x^2/289) + (y^2/225) = 1.[/tex]
This equation represents an ellipse centered at the point (0, 0) with a semi-major axis of length 17 and a semi-minor axis of length 15.
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Answer:
The equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is
To determine the equation of an ellipse given its foci and the length of its major axis, we need to use the standard form equation for an ellipse. The standard form equation for an ellipse centered at the origin is:
where 'a' represents the semi-major axis and 'b' represents the semi-minor axis.
In this case, we know that the distance between the foci is equal to 2a, which means a = 34/2 = 17. The foci of the ellipse are given as (7, 17) and (7, -13). The foci lie on the major axis of the ellipse, and since their y-coordinates differ by 30 (17 - (-13) = 30), the length of the major axis is equal to 2b, which means b = 30/2 = 15.
Now we have the values of a and b, so we can substitute them into the standard form equation:
Simplifying further, we have:
Therefore, the equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is:
This equation represents an ellipse centered at the point (0, 0) with a semi-major axis of length 17 and a semi-minor axis of length 15.
A village P is 12 km from village Q. It takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6 km/h from P to Q and (6 + x) km/h back to P, find the value of x.
Answer:
Hope this helps and have a nice day
Step-by-step explanation:
To find the value of x, we can use the formula:
Time = Distance / Speed
Let's calculate the time taken to travel from Q to P and back to Q.
From Q to P:
Distance = 12 km
Speed = 6 km/h
Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours
From P to Q:
Distance = 12 km
Speed = (6 + x) km/h
Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h
Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:
Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours
According to the problem, the total time is the sum of the time from Q to P and from P to Q:
Total time = Time taken from Q to P + Time taken from P to Q
Substituting the values:
10/3 hours = 2 hours + 12 km / (6 + x) km/h
Simplifying the equation:
10/3 = 2 + 12 / (6 + x)
Multiply both sides by (6 + x) to eliminate the denominator:
10(6 + x) = 2(6 + x) + 12
60 + 10x = 12 + 2x + 12
Collecting like terms:
8x = 24
Dividing both sides by 8:
x = 3
Therefore, the value of x is 3.
Answer:
x = 3
Step-by-step explanation:
speed = distance / time
time = distance / speed
Total time from P to Q to P:
T = 3h 20min
P to Q :
s = 6 km/h
d = 12 km
t = d/s
= 12/6
t = 2 h
time remaining t₁ = T - t
= 3h 20min - 2h
= 1 hr 20 min
= 60 + 20 min
= 80 min
t₁ = 80/60 hr
Q to P:
d₁ = 12km
t₁ = 80/60 hr
s₁ = d/t₁
[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]
= 9
s₁ = 9 km/h
From question, s₁ = (6 + x)km/h
⇒ 6 + x = 9
⇒ x = 3
Assume that each circle shown below represents one unit.express the shaded amount as a single fraction and as a mixed number
One fraction :
Mixed number:
The shape is represented as below
As one fraction = 9/4As a mixed number = 2 1/4How to represent the figure as a fractionThe figure is of three shapes, the firs two are whole numbers then the last is a fraction.
Adding them results to
shape 1 + shape 2 + shape 3
1 + 1 + 1/4
As one fraction
= 9/4
as a mixed number
= 2 1/4
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Please help me with this question
An estimate for the mean is 47.6 kg.
How to calculate the mean for the set of data?In Mathematics and Geometry, the mean for this set of data can be calculated by using the following formula:
Mean = [F(x)]/n
Cumulative frequency = 10 + 7 + 2 + 8 + 3
Cumulative frequency = 30
For the total number of data based on the frequency, we have;
Total weight, F(x) = 10(40) + 7(52.5) + 2(65) + 8(77.5) + 3(90)
Total weight, F(x) = 40 + 367.5 + 130 + 620 + 270
Total weight, F(x) = 1427.5
Now, we can calculate the mean weight as follows;
Mean = 1427.5/30
Mean = 47.6 kg.
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Trent has an 8-foot tall tent in the shape of square based pyramid with a base length of 14 feet. If one bottle of waterproof spray covers 76 square feet, how many bottles will he need to waterproof his tent.
Trent will need approximately 2.86 bottles of waterproof spray to cover his tent.
To calculate the number of bottles of waterproof spray Trent will need to cover his tent, we first need to find the surface area of the tent.
The surface area of a square-based pyramid is given by the formula:
Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)
The base of the pyramid is a square with a side length of 14 feet, so the base area is:
Base Area = (Side Length)^2 = 14^2 = 196 square feet
To find the slant height of the pyramid, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by one side of the base, the height of the pyramid, and the slant height. The height of the pyramid is given as 8 feet, and half the length of the base is 7 feet.
Using the Pythagorean theorem:
[tex]Slant Height^2 = (Half Base Length)^2 + Height^2[/tex]
[tex]Slant Height^2 = 7^2 + 8^2Slant Height^2 = 49 + 64Slant Height^2 = 113Slant Height ≈ √113 ≈ 10.63 feet[/tex]
Now we can calculate the surface area of the tent:
Surface Area = 196 + (0.5 x 4 x 10.63)
Surface Area = 196 + (2 x 10.63)
Surface Area = 196 + 21.26
Surface Area ≈ 217.26 square feet
Since each bottle of waterproof spray covers 76 square feet, we can divide the total surface area of the tent by the coverage of each bottle to find the number of bottles needed:
Number of Bottles = Surface Area / Coverage per Bottle
Number of Bottles = 217.26 / 76
Number of Bottles ≈ 2.86
Therefore, Trent will need approximately 2.86 bottles of waterproof spray to cover his tent. Since we can't have a fraction of a bottle, he will need to round up to the nearest whole number. Therefore, Trent will need 3 bottles of waterproof spray to fully waterproof his tent.
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Rotate point (-3, 2) about the origin 180 degrees clockwise. Where will the new point be?
Answer: the answer is (3,-2)
Step-by-step explanation: when you rotate a point about the origin 180 degrees clockwise, (x,y) turns into (-x,-y)
therefore
(-3,2) becomes (3,-2)
I'm pretty sure
A company Charting its profits notices that the relationship between the number of units sold,x, and the profit,P, is a linear. If 170 units sold results in $20 profit and 220 units sold results in $2820 profit, write the profit function for this company.
P=
Find the marginal profit
$
Step-by-step explanation:
a linear relationship or function is described in general as
y = f(x) = ax + b
Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".
here f(x) is P(x) :
P(x) = ax + b
now we are using both given points (ordered pairs) to calculate a and b :
20 = a×170 + b
2820 = a×220 + b
to eliminate first one variable we subtract equation 1 from equation 2 :
2800 = a×50
a = 2800/50 = 280/5 = 56
now, we use that in any of the 2 original equations to get b :
20 = 56×170 + b
b = 20 - 56×170 = 20 - 9520 = -9500
so,
P(x) = 56x - 9500
what best describes the relationship between the computed mean of 52.4 and the actual mean of 52.7
The computed mean of 52.4 and the actual mean of 52.7 suggest a close relationship in terms of central tendency.
A computed mean is a statistical measure calculated by summing up a set of values and dividing by the number of observations. In this case, the computed mean of 52.4 implies that when the values are averaged, the result is 52.4.
The actual mean of 52.7 refers to the true average of the population or data set being analyzed. Since it is higher than the computed mean, it indicates that the sample used for computation might have slightly underestimated the true population mean.
However, the difference between the computed mean and the actual mean is relatively small, with only a 0.3 unit discrepancy.
Given the proximity of these two values, it suggests that the computed mean is a reasonably accurate estimate of the actual mean.
However, it's important to note that without additional information, such as the sample size or the variability of the data, it is difficult to draw definitive conclusions about the relationship between the computed mean and the actual mean.
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Given ABCD, what is the measure of
145
A. 90°
B. 35°
C. 10°
D. 145°
E. 55°
F. 235°
Answer: D. 145°
Step-by-step explanation:
Since it is a parallelogram given by the symbol, then angle B is equal to angle D which is 145°.
James wants to have earned $6,180 amount of interest in 28 years. Currently he finds
that his annual interest rate is 6.12%. Calculate how much money James needs to invest
as his principal in order to achieve this goal.
Answer:
$3606.44
Step-by-step explanation:
The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.
To do this, we need to use the formula for simple interest:
[tex]\boxed{I = \frac{P \times R \times T}{100}}[/tex],
where:
I = interest earned
P = principal invested
R = annual interest rate
T = time
By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:
[tex]6180 = \frac{P \times 6.12 \times 28}{100}[/tex]
⇒ [tex]6180 \times 100 = P \times 171.36[/tex] [Multiplying both sides by 100]
⇒ [tex]P = \frac{6180 \times 100}{171.36}[/tex] [Dividing both sides of the equation by 171.36]
⇒ [tex]P = \bf 3606.44[/tex]
Therefore, James needs to invest $3606.44.
45% of the Walton High School student body are male. 90% of Walton females love math, while only 60% of the males love math. What percentage of the student body loves math?
Approximately 76.5% of the student body at Walton High School loves math.
To determine the percentage of the student body that loves math, we need to consider the proportions of males and females in the Walton High School student body and their respective percentages of loving math.
Given that 45% of the student body are males, we can deduce that 55% are females (since the total percentage must add up to 100%). Now let's calculate the percentage of the student body that loves math:
For the females:
55% of the student body are females.
90% of the females love math.
So, the percentage of females who love math is 55% * 90% = 49.5% of the student body.
For the males:
45% of the student body are males.
60% of the males love math.
So, the percentage of males who love math is 45% * 60% = 27% of the student body.
To find the total percentage of the student body that loves math, we add the percentages of females who love math and males who love math:
49.5% + 27% = 76.5%
As a result, 76.5% of Walton High School's student body enjoys maths.
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[tex]\sqrt{x+7}-1=x[/tex]
Answer:
x = 2
Step-by-step explanation:
Pre-SolvingWe are given the following equation:
[tex]\sqrt{x+7} -1=x[/tex], which we want to solve for x.
To do this, we should isolate the square root on one side, then square both sides. We can then solve the equation as normal, but then we have to check the domain in the end for any extraneous solutions.
SolvingStart by adding 1 to both sides.
[tex]\sqrt{x+7} -1=x[/tex]
+1 +1
________________________
[tex]\sqrt{x+7} = x+1[/tex]
Now, square both sides.
[tex](\sqrt{x+7} )^2= (x+1)^2[/tex]
We get:
x + 7 = x² + 2x + 1
Subtract x + 7 from both sides.
x + 7 = x² + 2x + 1
-(x+7) -(x+7)
________________________
0 = x² + x - 6
This can be factored to become:
0 = (x+3)(x-2)
Solve:
x+3 = 0
x = -3
x-2 = 0
x = 2
We get x = -3 and x = 2. However, we must check the domain.
DomainSubstitute -3 as x and 2 as x into the original equation.
We get:
[tex]\sqrt{-3+7} -1 = -3[/tex]
[tex]\sqrt{4} -1 = -3[/tex]
2 - 1 = -3
-1 = -3
This is an untrue statement, so x = -3 is an extraneous solution.
We also get:
[tex]\sqrt{2+7} -1 = 2[/tex]
[tex]\sqrt{9}-1=2[/tex]
3 - 1 = 2
2 = 2
This is a true statement, so x = 2 is a real solution.
Our only answer is x = 2.
Given the function f(x) = 4 – 2x, find f(3r – 1).
Answer:
f(3r - 1) = -6r + 6
Step-by-step explanation:
To find f(3r - 1), we substitute 3r - 1 for x in the expression for f(x) and simplify:
f(x) = 4 - 2x
f(3r - 1) = 4 - 2(3r - 1)
= 4 - 6r + 2
= -6r + 6
So, f(3r - 1) = -6r + 6.
Solve using inverse (matrix) method
5x - 4y + z = 12
x + 7y-z = -9
2x+3y + 3z = 8
The solution to the system of equations using the inverse matrix method is x = -1, y = 2, z = 3.
To solve the system of equations using the inverse matrix method, we need to represent the system in matrix form.
The given system of equations can be written as:
| 5 -4 1 | | x | = | 12 |
| 1 7 -1 | [tex]\times[/tex]| y | = | -9 |
| 2 3 3 | | z | | 8 |
Let's denote the coefficient matrix on the left side as A, the variable matrix as X, and the constant matrix as B.
Then the equation can be written as AX = B.
Now, to solve for X, we need to find the inverse of matrix A.
If A is invertible, we can calculate X as [tex]X = A^{(-1)} \times B.[/tex]
To find the inverse of matrix A, we can use the formula:
[tex]A^{(-1)} = (1 / det(A)) \times adj(A)[/tex]
Where det(A) is the determinant of A and adj(A) is the adjugate of A.
Calculating the determinant of A:
[tex]det(A) = 5 \times (7 \times 3 - (-1) \times 3) - (-4) \times (1 \times 3 - (-1) \times 2) + 1 \times (1 \times (-1) - 7\times 2)[/tex]
= 15 + 10 + (-13)
= 12.
Next, we need to find the adjugate of A, which is obtained by taking the transpose of the cofactor matrix of A.
Cofactor matrix of A:
| (73-(-1)3) -(13-(-1)2) (1(-1)-72) |
| (-(53-(-1)2) (53-12) (5[tex]\times[/tex] (-1)-(-1)2) |
| ((5(-1)-72) (-(5(-1)-12) (57-(-1)[tex]\times[/tex](-1)) |
Transpose of the cofactor matrix:
| 20 -7 -19 |
| 13 13 -3 |
| -19 13 36 |
Finally, we can calculate the inverse of A:
A^(-1) = (1 / det(A)) [tex]\times[/tex] adj(A)
= (1 / 12) [tex]\times[/tex] | 20 -7 -19 |
| 13 13 -3 |
| -19 13 36 |
Multiplying[tex]A^{(-1)[/tex] with B, we can solve for X:
[tex]X = A^{(-1)}\times B[/tex]
= | 20 -7 -19 | | 12 |
| 13 13 -3 | [tex]\times[/tex] | -9 |
| -19 13 36 | | 8 |
Performing the matrix multiplication, we can find the values of x, y, and z.
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Suppose a finite population has 6 items and 2 items are selected at random without replacement,then all possible samples will be:
Select one:
a. 15
b. 2
c. 36
d. 6
e. 12
Note: Answer D is NOT the correct answer. Please find the correct answer. Any answer without justification will be rejected automatically.
When 2 items are selected without replacement from a population of 6 items, there are 15 possible samples that can be formed. Option A.
To determine the number of possible samples when 2 items are selected at random without replacement from a population of 6 items, we can use the concept of combinations.
The number of combinations of selecting k items from a set of n items is given by the formula C(n, k) = n! / (k! * (n-k)!), where n! represents the factorial of n.
In this case, we have a population of 6 items and we want to select 2 items. Therefore, the number of possible samples can be calculated as:
C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / (2 * 1) = 15. Option A is correct.
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What else would need to be congruent to show that ABC=AXYZ by SAS?
A
B
OA. ZB=LY
B. BC = YZ
OC. C= LZ
OD. AC = XZ
с
X
Z
Given:
AB XY
BC=YZ
What is needed to be congruent to show that ABC=AXYZ is AC ≅ XZ. option D
How to determine the statementGiven that in ΔABC and ΔXYZ, ∠X ≅ ∠A and ∠Z ≅ ∠C.
We are to select the correct condition that we will need to show that the triangles ABC and XYZ are congruent to each other by ASA rule..
ASA Congruence Theorem: Two triangles are said to be congruent if two angles and the side lying between them of one triangle are congruent to the corresponding two angles and the side between them of the second triangle.
In ΔABC, side between ∠A and ∠C is AC,
in ΔXYZ, side between ∠X and ∠Z is XZ.
Therefore, for the triangles to be congruent by ASA rule, we must have AC ≅ XZ.
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The pyramid and prism above have the same triangular base and height. The volume of the pyramid is 18 cubic inches. What is the volume of the prism?
A. 36 cubic inches
B. 72 cubic inches
C. 6 cubic inches
D. 54 cubic inches
which of the following are like radicals? Check all
of the boxes that apply.
3x√√xy
-12x√√xy
-2x√√xj
x-√4x2²
-x√x²y
2√xy
Answer:
the first 2
Step-by-step explanation:
let me know if it is wrong