(b) Consider the heat conduction problem
Uxx = ut, 0 < x < 30, t > 0,
u(0,t) = 20, u(30,t) = 50, u(x, 0) = 60- 2x, 0 < x < 30. t > 0,
Find the steady-state temperature distribution and the boundary value problem that
determines the transient distribution.

Answer:

  C.  75°

Step-by-step explanation:

You want the angle marked ∠1 in the trapezoid shown.

Where a transversal crosses parallel lines, same-side interior angles are supplementary. In this trapezoid, this means the angles at the right side of the figure are supplementary:

  ∠1 + 105° = 180°

  ∠1 = 75° . . . . . . . . . . . . subtract 105°

__

Additional comment

The given relation also means that the unmarked angle is supplementary to the one marked 50°. The unmarked angle will be 130°.

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The Fourier series of f(t) = 31² is a₀ = 31² and all other coefficients are zero.

For (i)[tex]2x^5[/tex] - 5x³ + 7: even, (ii) x³ + x⁴: odd.

The Fourier series of f(x) = x is Σ(bₙsin(nπx/L)), where b₁ = 4L/π.

To find the Fourier series of the periodic function f(t) = 31² over the interval -1 ≤ t ≤ 1, we need to determine the coefficients of its Fourier series representation. Since f(t) is a constant function, all the coefficients except for the DC component will be zero. The DC component (a₀) is given by the average value of f(t) over one period, which is equal to the constant value of f(t). In this case, a₀ = 31².

For the functions (i)[tex]2x^5[/tex] - 5x³ + 7 and (ii) x³ + x⁴, we can determine their symmetry by examining their even and odd components. A function is even if f(-x) = f(x) and odd if f(-x) = -f(x).

(i) For[tex]2x^5[/tex] - 5x³ + 7, we observe that the even powers of x (x⁰, x², x⁴) are present, while the odd powers (x¹, x³, x⁵) are absent. Thus, the function is even.

(ii) For x³ + x⁴, both even and odd powers of x are present. By testing f(-x), we find that f(-x) = -x³ + x⁴ = -(x³ - x⁴) = -f(x). Hence, the function is odd.

For the function f(x) = x over the interval -L ≤ x ≤ L, we can determine its Fourier series by finding the coefficients of its sine terms. The Fourier series representation of f(x) is given by f(x) = a₀/2 + Σ(aₙcos(nπx/L) + bₙsin(nπx/L)), where a₀ = 0 and aₙ = 0 for all n > 0.

Since f(x) = x is an odd function, only the sine terms will be present in its Fourier series. The coefficient b₁ can be determined by integrating f(x) multiplied by sin(πx/L) over the interval -L to L and then dividing by L.

The Fourier series for f(x) = x over -L ≤ x ≤ L is given by f(x) = Σ(bₙsin(nπx/L)), where b₁ = 4L/π.

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The graph of sinusoidal functions f (x) and g (x) are shown in graph.

And, the transformation of each function is shown below.

We have,

Two sinusoidal functions,

a. f(x) = - 3 cos(45(x - 2°)) + 5

b. g(x) = 2.5 sin(- 3(x+90° )) - 1

Now, Let's break down the transformations for each function:

a. For the function f(x) = -3⋅cos(45(x-2°)) + 5:

The coefficient in front of the cosine function, -3, represents the amplitude.

It determines the vertical stretching or compression of the graph. In this case, the amplitude is 3, but since it is negative, the graph will be reflected across the x-axis.

And, The period of the cosine function is normally 2π, but in this case, we have an additional factor of 45 in front of the x.

This means the period is shortened by a factor of 45, resulting in a period of 2π/45.

And, The phase shift is determined by the constant inside the parentheses, which is -2° in this case.

A positive value would shift the graph to the right, and a negative value shifts it to the left.

So, the graph is shifted 2° to the right.

Since, The constant term at the end, +5, represents the vertical shift of the graph. In this case, the graph is shifted 5 units up.

b. For the function g(x) = 2.5⋅sin(-3(x+90°)) - 1:

Here, The coefficient in front of the sine function, 2.5, represents the amplitude. It determines the vertical stretching or compression of the graph. In this case, the amplitude is 2.5, and since it is positive, there is no reflection across the x-axis.

Period: The period of the sine function is normally 2π, but in this case, we have an additional factor of -3 in front of the x.

This means the period is shortened by a factor of 3, resulting in a period of 2π/3.

Phase shift: The phase shift is determined by the constant inside the parentheses, which is +90° in this case.

A positive value would shift the graph to the left, and a negative value shifts it to the right.

So, the graph is shifted 90° to the left.

Vertical shift: The constant term at the end, -1, represents the vertical shift of the graph.

In this case, the graph is shifted 1 unit down.

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The instantaneous growth rate of an investment represents the rate at which its value is increasing at any given moment. In this case, the interest rate is 5%, which means that the investment grows by 5% each year.

In the first step, to calculate the instantaneous growth rate, we simply take the given interest rate, which is 5%.

In the second step, to find the amount of the investment after 5 years when compounded continuously, we use the continuous compounding formula: A = P * e^(rt), where A is the final amount, P is the principal (initial investment), e is the base of the natural logarithm, r is the interest rate, and t is the time in years. Plugging in the values, we have A = 12000 * e^(0.05 * 5) ≈ $16,283.19.

In the third step, to find the amount of the investment after 5 years when compounded quarterly, we use the compound interest formula: A = P * (1 + r/n)^(nt), where n is the number of compounding periods per year. In this case, n is 4 since the investment is compounded quarterly. Plugging in the values, we have A = 12000 * (1 + 0.05/4)^(4 * 5) ≈ $16,209.62.

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The given sequence an = 1 (n+4)! (4n+ 1)! is decreasing and bounded. Option 2 is the correct answer.

To determine whether the sequence

[tex]an = 1/(n+4)!(4n+1)![/tex]

is increasing, decreasing, or neither, we can look at the ratio of consecutive terms:

Thus,

[tex]a(n+1)/an = [1/(n+5)!(4n+5)!] / [1/(n+4)!(4n+1)!] \\

= [(n+4)!(4n+1)!] / [(n+5)!(4n+5)!] \\

= (4n+1)/(4n+5)[/tex]

The ratio of consecutive terms is a decreasing function of n, since (4n+1)/(4n+5) < 1 for all n.

Hence, the sequence is decreasing.

To determine whether the sequence is bounded, we need to find an upper bound and a lower bound for the sequence.

Note that all terms of the sequence are positive, since the factorials and the denominator of each term are positive.

We can use the inequality

[tex](4n+1)! < (4n+1)^{4n+1/2}[/tex]

to obtain an upper bound for the sequence:

[tex]an < 1/(n+4)!(4n+1)! \\

< 1/[(n+4)/(4n+1)^{4n+1/2}] \\

< 1/[(1/4)(n^{1/2})][/tex]

Therefore, the sequence is bounded above by

[tex]4n^{1/2}.[/tex]

Therefore, the sequence is decreasing and bounded.

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(a) The permutations of the set {1, 2, 3, 4} corresponding to r and s are:

r = (1 2 3 4)

s = (1 4)(2 3)

(b) Using the permutations from part (a), we can show that rs = sr^3:

rs = (1 2 3 4)(1 4)(2 3)

= (1 2 3 4)(1 4 2 3)

= (1 4 2 3)

sr^3 = (1 4)(2 3)(1 2 3 4)

= (1 4)(2 3 1 4)

= (1 4 2 3)

Therefore, rs = sr^3.

(a) The permutation r corresponds to a 90-degree clockwise rotation of the square, which can be represented as (1 2 3 4), indicating that vertex 1 is mapped to vertex 2, vertex 2 is mapped to vertex 3, and so on. The permutation s corresponds to a reflection about a vertical axis, which swaps the positions of vertices 1 and 4, as well as vertices 2 and 3. Therefore, it can be represented as (1 4)(2 3), indicating that vertex 1 is swapped with vertex 4, and vertex 2 is swapped with vertex 3. (b) To show that rs = sr^3, we substitute the permutations from part (a) into the expression: rs = (1 2 3 4)(1 4)(2 3)

= (1 2 3 4)(1 4 2 3)

= (1 4 2 3)

Similarly, we evaluate sr^3:

sr^3 = (1 4)(2 3)(1 2 3 4)

= (1 4)(2 3 1 4)

= (1 4 2 3)

By comparing the results, we can see that rs and sr^3 are equal. Hence, we have shown that rs = sr^3 using the permutations obtained in part (a).

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Solving given equations, we get line of intersection as  t = -11/4, t = -1, and t = 1/4, respectively. The point of intersection between the given lines is (-8, 2, 0). The point of intersection between the two planes is (2, 2, 86/65).

(10.2) To find the line of intersection between the lines, let's set up the equations for the two lines:

Line 1: r1 = <3, -1, 2> + t<1, 1, -1>

Line 2: r2 = <-8, 2, 0> + t<-3, 2, -7>

Now, we equate the two lines to find the point of intersection:

<3, -1, 2> + t<1, 1, -1> = <-8, 2, 0> + t<-3, 2, -7>

By comparing the corresponding components, we get:

3 + t = -8 - 3t   [x-component]

-1 + t = 2 + 2t   [y-component]

2 - t = 0 - 7t    [z-component]

Simplifying these equations, we find:

4t = -11   [from the x-component equation]

-3t = 3     [from the y-component equation]

8t = 2      [from the z-component equation]

Solving these equations, we get t = -11/4, t = -1, and t = 1/4, respectively.

To find the point of intersection, substitute the values of t back into any of the original equations. Taking the y-component equation as an example, we have:

-1 + t = 2 + 2t

Substituting t = -1, we find y = 2.

Therefore, the point of intersection between the given lines is (-8, 2, 0).

(10.3) Let's solve for the point of intersection between the two given planes:

Plane 1: -5x + y - 2z = 3

Plane 2: 2x - 3y + 5z = -7

To find the point of intersection, we need to solve this system of equations simultaneously. We can use the method of substitution or elimination to find the solution.

Let's use the method of elimination:

Multiply the first equation by 2 and the second equation by -5 to eliminate the x term:

-10x + 2y - 4z = 6

-10x + 15y - 25z = 35

Now, subtract the second equation from the first equation:

0x - 13y + 21z = -29

To simplify the equation, divide through by -13:

y - (21/13)z = 29/13

Now, let's solve for y in terms of z:

y = (21/13)z + 29/13

We still need another equation to find the values of z and y. Let's use the y-component equation from the second plane:

y - 3 = -5s

Substituting y = (21/13)z + 29/13, we have:

(21/13)z + 29/13 - 3 = -5s

Simplifying, we get:

(21/13)z - (34/13) = -5s

Now, we can equate the z-components of the two equations:

(21/13)z - (34/13) = 2z + 4

Simplifying further, we have:

(21/13)z - 2z = (34/13) + 4

(5/13)z = (34/13) + 4

(5/13)z = (34 + 52)/13

(5/13)z =

86/13

Solving for z, we find z = 86/65.

Substituting this value back into the y-component equation, we can find the value of y:

y = (21/13)(86/65) + 29/13

Simplifying, we have: y = 2

Therefore, the point of intersection between the two planes is (2, 2, 86/65).

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a. The truck rental cost when you drive 85 miles is  $85.7.

b. The number of miles driven when the cost is $65.96 is 0.42x.

a. To find the truck rental cost when driving 85 miles, we can substitute the value of x into the given function.

f(x) = 0.42x + 50

Substituting x = 85:

f(85) = 0.42(85) + 50

= 35.7 + 50

= 85.7

Therefore, the truck rental cost when driving 85 miles is $85.70.

b. To determine the number of miles driven when the cost is $65.96, we can set up an equation using the given function.

f(x) = 0.42x + 50

Substituting f(x) = 65.96:

65.96 = 0.42x + 50

Subtracting 50 from both sides:

65.96 - 50 = 0.42x

15.96 = 0.42x

To isolate x, we divide both sides by 0.42:

15.96 / 0.42 = x

38 = x

Therefore, the number of miles driven when the cost is $65.96 is 38 miles.

In summary, when driving 85 miles, the truck rental cost is $85.70, and when the cost is $65.96, the number of miles driven is 38 miles.

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A suitable form of Y(t) is [tex]$$Y(t) = c_1 e^{2\sqrt2t} + c_2 e^{-2\sqrt2t} + c_3 \cos 2t + c_4 \sin 2t + At^2 + Bt + C + D\sin t + E\cos t.$$[/tex]

The method of undetermined coefficients is an effective way of finding the particular solution to the differential equations when the right-hand side is a sum or a constant multiple of exponentials, sine, cosine, and polynomial functions.

Let's solve the given equation using the method of undetermined coefficients.

[tex]$$y^{4} − 4y''''- 16y'' + 64y' = t^2-3+t\sin t$$[/tex]

The characteristic equation is [tex]$r^4 -4r^2 - 16r +64 =0.$[/tex]

Factorizing it, we get

[tex]$(r^2 -8)(r^2 +4) = 0$[/tex]

So the roots are [tex]$r_1 = 2\sqrt2, r_2 = -2\sqrt2, r_3 = 2i$[/tex] and [tex]$r_4 = -2i$[/tex]

Thus, the homogeneous solution is given by

[tex]$$y_h(t) = c_1 e^{2\sqrt2t} + c_2 e^{-2\sqrt2t} + c_3 \cos 2t + c_4 \sin 2t$$[/tex]

Now, let's find a particular solution using the method of undetermined coefficients. A suitable form of the particular solution is:

[tex]$$y_p(t) = At^2 + Bt + C + D\sin t + E\cos t.$$[/tex]

Taking the derivatives of [tex]$y_p(t)$[/tex] , we have

[tex]$$y_p'(t) = 2At + B + D\cos t - E\sin t$$$$y_p''(t) = 2A - D\sin t - E\cos t$$$$y_p'''(t) = D\cos t - E\sin t$$$$y_p''''(t) = -D\sin t - E\cos t$$[/tex]

Substituting the forms of[tex]$y_p(t)$, $y_p'(t)$, $y_p''(t)$, $y_p'''(t)$ and $y_p''''(t)$[/tex] in the given differential equation,

we get[tex]$$(-D\sin t - E\cos t) - 4(D\cos t - E\sin t) - 16(2A - D\sin t - E\cos t) + 64(2At + B + C + D\sin t + E\cos t) = t^2 - 3 + t\sin t$$[/tex]

Simplifying the above equation, we get

[tex]$$(-192A + 64B - 18)\cos t + (192A + 64B - 17)\sin t + 256At^2 + 16t^2 - 12t - 7=0.$$[/tex]

Now, we can equate the coefficients of the terms [tex]$\sin t$, $\cos t$, $t^2$, $t$[/tex], and the constant on both sides of the equation to solve for the constants A B C D & E

Therefore, a suitable form of

[tex]Y(t) is$$Y(t) = c_1 e^{2\sqrt2t} + c_2 e^{-2\sqrt2t} + c_3 \cos 2t + c_4 \sin 2t + At^2 + Bt + C + D\sin t + E\cos t.$$[/tex]

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Coupon STRIPS can be created from the given T-bond by removing the coupon payments from the bond and selling them as individual securities. Let's calculate how many coupon STRIPS can be created from this T-bond.

The bond has a 5% coupon, which means it will pay $5 million in interest every year. Over a period of 29 years, the total interest payments would be $5 million x 29 years = $145 million.

The par value of the bond is $100 million. After deducting the interest payments of $145 million, the remaining principal value is $100 million - $145 million = -$45 million.

Since there is a negative principal value, we cannot create any principal STRIPS from this bond. However, we can create coupon STRIPS equal to the number of coupon payments that will be made over the remaining life of the bond.

Therefore, we can create 29 coupon STRIPS of $5 million each from this T-bond. These coupon STRIPS will be sold separately and will not include the principal repayment of the bond.

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The equation of the least-squares line that best fits the given data points is y = 2 + (2/3)x.

To find the equation of the least-squares line that best fits the given data points, we can use the method of least squares to minimize the sum of the squared differences between the actual y-values and the predicted y-values on the line.

Calculate the mean of the x-values and the mean of the y-values.

[tex]\bar x[/tex] = (0 + 1 + 2 + 3) / 4 = 1.5

[tex]\bar y[/tex]= (2 + 2 + 5 + 5) / 4 = 3.5

Calculate the deviations from the means for both x and y.

x₁ = 0 - 1.5 = -1.5

x₂ = 1 - 1.5 = -0.5

x₃ = 2 - 1.5 = 0.5

x₄ = 3 - 1.5 = 1.5

y₁ = 2 - 3.5 = -1.5

y₂ = 2 - 3.5 = -1.5

y₃ = 5 - 3.5 = 1.5

y₄ = 5 - 3.5 = 1.5

Calculate the sum of the products of the deviations from the means.

Σ(xᵢ * yᵢ) = (-1.5 * -1.5) + (-0.5 * -1.5) + (0.5 * 1.5) + (1.5 * 1.5) = 4

Calculate the sum of the squared deviations of x.

Σ(xᵢ²) = (-1.5)² + (-0.5)² + (0.5)² + (1.5)² = 6

Calculate the least-squares slope (B₁) using the formula:

B₁ = Σ(xᵢ * yᵢ) / Σ(xᵢ²) = 4 / 6 = 2/3

Calculate the y-intercept (Bo) using the formula:

Bo = [tex]\bar y[/tex] - B₁ * [tex]\bar x[/tex] = 3.5 - (2/3) * 1.5 = 2

Therefore, the equation of the least-squares line that best fits the given data points is y = 2 + (2/3)x.

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Consider the system x'=8y+x+12 y'=x−y+12t

A. The eigenvalues of the matrix A are the solutions to the characteristic equation λ³ - 12λ² + 25λ - 12 = 0.

B. The eigenvectors corresponding to the eigenvalues can be found by solving the equation (A - λI)v = 0, where v is the eigenvector.

C. The general solution of the homogeneous system can be expressed as a linear combination of the eigenvectors corresponding to the eigenvalues.

D. To find the particular solution of the non-homogeneous system, substitute the given values into the system of equations and solve for the variables.

E. The general solution of the non-homogeneous system is the sum of the general solution of the homogeneous system and the particular solution of the non-homogeneous system.

F. The behavior of the system as t approaches infinity depends on the eigenvalues and their corresponding eigenvectors. It can be determined by analyzing the values and properties of the eigenvalues, such as whether they are positive, negative, or complex, and considering the corresponding eigenvectors.

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The statement can be translated into symbolic notation as follows:

S(x): x is a student.

C(x): x takes Chemistry.

M(x): x passed Math.

E(x): x has an exam this week.

m: Mariam

Symbolic notation:

S(m) ∧ C(m) → E(m)

The given statement is translated into symbolic notation using predicate logic. In the notation, S(x) represents "x is a student," C(x) represents "x takes Chemistry," M(x) represents "x passed Math," E(x) represents "x has an exam this week," and m represents Mariam.

The translated statement S(m) ∧ C(m) → E(m) represents the logical implication that if Mariam is a student and Mariam takes Chemistry, then Mariam has an exam this week.

To determine the validity of the argument, we need to assess whether the logical implication holds true in all cases. If it does, the argument is considered valid.

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Duffer's Delite per unit contribution margin, including cannibalization as a variable cost, is $2.33.

The per unit contribution margin for Duffer's Delite can be calculated by subtracting the variable manufacturing cost and the cannibalization cost from the selling price. The variable manufacturing cost of Duffer's Delite is $5 per dozen, which translates to $0.42 per unit (5/12). The cannibalization cost is equal to the margin per unit of the StraightShot golf balls, which is $8 per dozen or $0.67 per unit (8/12). Therefore, the per unit contribution margin for Duffer's Delite is $12 - $0.42 - $0.67 = $10.91 - $1.09 = $9.82. However, since the per unit contribution margin is calculated based on one unit (12 golf balls), we need to divide it by 12 to get the per unit contribution margin for a single golf ball, which is $9.82/12 = $0.82. Finally, to account for the cannibalization cost, we need to subtract the cannibalization rate of 0.18 (as calculated previously) multiplied by the per unit contribution margin of the StraightShot golf balls ($0.82) from the per unit contribution margin of Duffer's Delite. Therefore, the final per unit contribution margin for Duffer's Delite, including cannibalization, is $0.82 - (0.18 * $0.82) = $0.82 - $0.1476 = $0.6724, which can be rounded to $0.67 or $2.33 per dozen.

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To solve the given questions, let's analyze each part one by one:

a) The y-intercept is (0, 0).

Find the x- and y-intercepts of y = f(x):

The x-intercepts are the points where the graph of the function intersects the x-axis, meaning the y-coordinate is zero. To find the x-intercepts, set y = 0 and solve for x:

0 = 4x²(x² - 9)

This equation can be factored as:

0 = 4x²(x + 3)(x - 3)

From this factorization, we can see that there are three possible solutions for x:

x = 0 (gives the x-intercept at the origin, (0, 0))

x = -3 (gives an x-intercept at (-3, 0))

x = 3 (gives an x-intercept at (3, 0))

The y-intercept is the point where the graph intersects the y-axis, meaning the x-coordinate is zero. To find the y-intercept, substitute x = 0 into the equation:

y = 4(0)²(0² - 9)

y = 4(0)(-9)

y = 0

Therefore, the y-intercept is (0, 0).

b) Find the equation of all vertical asymptotes (if they exist):

Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a particular value. To find vertical asymptotes, we need to check where the function is undefined.

In this case, the function is undefined when the denominator of a fraction is equal to zero. The denominator in our case is (x² - 9), so we set it equal to zero:

x² - 9 = 0

This equation can be factored as the difference of squares:

(x - 3)(x + 3) = 0

From this factorization, we find that x = 3 and x = -3 are the values that make the denominator zero. These values represent vertical asymptotes.

Therefore, the equations of the vertical asymptotes are x = 3 and x = -3.

c) Find the equation of all horizontal asymptotes (if they exist):

To determine horizontal asymptotes, we need to analyze the behavior of the function as x approaches positive or negative infinity.

Given that the highest power of x in the numerator and denominator is the same (both are x²), we can compare their coefficients to find horizontal asymptotes. In this case, the coefficient of x² in the numerator is 4, and the coefficient of x² in the denominator is 1.

Since the coefficient of the highest power of x is greater in the numerator, there are no horizontal asymptotes in this case.

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The area of triangle ABC is 78units²

The space covered by the figure or any two-dimensional geometric shape, in a plane, is the area of the shape.

A triangle is a 3 sided polygon and it's area is expressed as;

A = 1/2bh

where b is the base and h is the height.

The area of triangle ABC = area of big triangle- area of the 2 small triangles+ area of square

Area of big triangle = 1/2 × 13 × 18

= 18 × 9

= 162

Area of small triangle = 1/2 × 8 × 6

= 24

area of small triangle = 1/2 × 12 × 5

= 30

area of rectangle = 5 × 6 = 30

= 24 + 30 +30 = 84

Therefore;

area of triangle ABC = 162 -( 84)

= 78 units²

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(a) "For all real numbers r, there exists a real number s such that s squared is equal to r."

(b) True - The statement holds true for all real numbers.

(a) The symbolic statement "Vr R, 3s R, s² = r" can be written in English as "For all real numbers r, there exists a real number s such that s squared is equal to r."

(b) The statement is true. It asserts that for any real number r, there exists a real number s such that s squared is equal to r. This is a true statement because for every positive real number r, we can find a positive real number s such that s squared equals r (e.g., s = √r). Similarly, for every negative real number r, we can find a negative real number s such that s squared equals r (e.g., s = -√r). Therefore, the statement holds true for all real numbers.

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The correct options to fill in the gaps are:

From the diagram given, we have that;

We are to show that the segment AB is congruent to DF

Also from the diagram

Similarly, given that:

AB = CE and CE = DF implies AB = DF by the Transitive Property of Equality.

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The complete question is in the attached figure below.

The value of h is -2. The phase shift is 2 units to the left.

Given function:

g(t)=f(t+2)

The general form of the function is

g(t) = f(t-h)

where h is the horizontal translation or phase shift in the function. The function g(t) is translated by 2 units in the left direction compared to f(t). Therefore the answer is that the value of h in the translation is -2.

The phase shift can be described as the transformation of the graph of a function in which the function is moved along the x-axis by a certain amount of units. The phrase used to describe this transformation is “units to the left” or “units to the right” depending on the direction of the transformation. In this case, the phase shift is towards the left of the graph by 2 units. The phrase used to describe the phase shift is “2 units to the left.”

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(a) P(0) = 9000, P(4) ≈ 23051.

(b) The population will reach 18,000 in approximately 5 years.

(a). To find the population at time t=0, we substitute t=0 into the population growth function:

P(0) = 9000(1.3)[tex]^0[/tex] = 9000

To find the population at time t=4, we substitute t=4 into the population growth function:

P(4) = 9000(1.3)[tex]^4[/tex] ≈ 23051

Therefore, the population at time t=0 is 9000 and the population at time t=4 is approximately 23051.

(b). To determine when the population will reach 18,000, we need to solve the equation:

18000 = 9000(1.3)[tex]^t[/tex]

Divide both sides of the equation by 9000:

2 = (1.3)[tex]^t[/tex]

To solve for t, we can take the logarithm of both sides using any base. Let's use the natural logarithm (ln):

ln(2) = ln((1.3)[tex]^t[/tex])

Using the logarithmic property of exponents, we can bring the exponent t down:

ln(2) = t * ln(1.3)

Now, divide both sides of the equation by ln(1.3) to isolate t:

t = ln(2) / ln(1.3) ≈ 5.11

Therefore, the population will reach 18,000 in approximately 5 years.

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Translating a sentence into a multi-step equation gives : 9 + (c/3) = 6.

1. Identify the unknown number and assign a variable to it.

In this case, the unknown number is represented by the variable c.

2. Translate the sentence into an equation.

The sentence states "Nine more than the quotient of a number and 3 is equal to 6." We can break this down into two parts. First, we have the quotient of a number and 3, which can be represented as c/3. Then, we add nine more to this quotient, resulting in 9 + (c/3). Finally, we set this expression equal to 6.

3. Justify the equation.

The equation 9 + (c/3) = 6 translates the sentence accurately. It states that when we divide a number (represented by c) by 3 and add 9 to the quotient, the result is 6. By solving this equation, we can find the value of c that satisfies the given condition.

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Answer: a. 3a^2 + 3

Step-by-step explanation: Use -a instead of x. -a * -a is a^2. Therefore the answer is positive which can only be choice a.

Empirical (E)

Theoretical (T)

Theoretical (T)

Theoretical (T)

The statement about COVID-19 deaths on a specific date is empirical because it is based on actual recorded data from worldometers.info.

The CDC's anticipation of a second wave of COVID cases during the flu season is a theoretical prediction. It is based on their understanding of viral transmission patterns and historical data from previous pandemics.

The statement about older adults and individuals with underlying medical conditions being at higher risk for serious complications from COVID-19 is a theoretical observation. It is based on analysis and studies conducted on the impact of the virus on different populations.

The prediction of lower enrollment in the upcoming semester by ASU is a theoretical projection. It is based on their analysis of various factors such as the ongoing pandemic's impact on student preferences and decisions.

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Example 1: y = x⁴ – x³ + x² – x + 1. Example 2: y = x⁴ + 6x² + 25.These polynomials have non-zero coefficients for the terms x³ and x², which means they cannot be expressed in the required form.

Quartic polynomials of the form y = a(x – h)⁴ + k cannot represent all quartic functions. Some quartic polynomials cannot be written in this form, for various reasons, including the presence of the term x³.Here are two examples of quartic polynomials that cannot be written in the form y = a(x – h)⁴ + k:

Example 1: y = x⁴ – x³ + x² – x + 1

This quartic polynomial does not have the same form as y = a(x – h)⁴ + k. It contains a term x³, which is not present in the given form. As a result, it cannot be written in the form y = a(x – h)⁴ + k.

Example 2: y = x⁴ + 6x² + 25

This quartic polynomial also does not have the same form as y = a(x – h)⁴ + k. It does not contain any linear or cubic terms, but it does have a quadratic term 6x². This means that it cannot be written in the form y = a(x – h)⁴ + k.Therefore, some quartic polynomials cannot be expressed in the form of y = a(x-h)⁴+k, as mentioned earlier. Two such examples are as follows:Example 1: y = x⁴ – x³ + x² – x + 1

Example 2: y = x⁴ + 6x² + 25

These polynomials have non-zero coefficients for the terms x³ and x², which means they cannot be expressed in the required form. These are the simplest examples of such polynomials; there may be more complicated ones as well, but the concept is the same.

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To find the median in a continuous series, the lower limit and frequency of the median class are important. The correct answer is option (b). For the given continuous data distribution, the value of Q3 is 30.

To find the median in a continuous series, the lower limit and frequency of the median class are important. Therefore, the correct answer is option (b).

To find Q3 in a continuous data distribution, we need to first find the median (Q2). The total frequency is 3+5+7+1 = 16, which is even. Therefore, the median is the average of the 8th and 9th values.

The 8th value is in the class 30-40, which has a cumulative frequency of 3+5 = 8. The lower limit of this class is 30. The class width is 10.

The 9th value is also in the class 30-40, so the median is in this class. The particular frequency of this class is 7. Therefore, the median is:

Q2 = lower limit of median class + [(n/2 - cumulative frequency of the class before median class) / particular frequency of median class] * class width

Q2 = 30 + [(8 - 8) / 7] * 10 = 30

To find Q3, we need to find the median of the upper half of the data. The upper half of the data consists of the classes 30-40 and 40-50. The total frequency of these classes is 7+1 = 8, which is even. Therefore, the median of the upper half is the average of the 4th and 5th values.

The 4th value is in the class 40-50, which has a cumulative frequency of 8. The lower limit of this class is 40. The class width is 10.

The 5th value is also in the class 40-50, so the median of the upper half is in this class. The particular frequency of this class is 1. Therefore, the median of the upper half is:

Q3 = lower limit of median class + [(n/2 - cumulative frequency of the class before median class) / particular frequency of median class] * class width

Q3 = 40 + [(4 - 8) / 1] * 10 = 0

Therefore, the correct answer is option (b): 30.

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Equation  [tex]c/E - 1/mc = 0[/tex]

Solve for E

E = mc

To solve the equation for E, we can start by isolating the term containing E on one side of the equation. Let's rearrange the equation step by step

c/E - 1/mc = 0

To eliminate the fraction, we can multiply every term by the common denominator, which is mcE

(mcE)(c/E) - (mcE)(1/mc) = (mcE)(0)

Simplifying

[tex]c^2 - E = 0[/tex]

Now, we can isolate E by moving c^2 to the other side of the equation

[tex]E = c^2[/tex]

The equation c/E - 1/mc = 0 can be solved to find that E is equal to c^2. This means that the value of E is the square of the constant c. By rearranging the original equation, we eliminate the fraction and simplify it to the form E = c^2. This result indicates that the value of E is solely determined by the square of c. Therefore, if we know the value of c, we can find E by squaring it.

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To maximize the sugar you can gain from street vendors and satisfy your sweet tooth, you can use the following expression:

m[i] = max(m[i-1] + s[i], s[i])

Here, m[i] represents the maximum sugar you can consume so far on the i-th vendor, and s[i] denotes the sugar content of the i-th vendor's offering.

The expression utilizes dynamic programming to calculate the maximum sugar consumption at each step. The variable m[i] stores the maximum sugar you can have up to the i-th vendor.

The expression considers two options: either including the sugar content of the current vendor (s[i]) or starting a new consumption from the current vendor.

To calculate m[i], we compare the sum of the maximum sugar consumption until the previous vendor (m[i-1]) and the sugar content of the current vendor (s[i]) with just the sugar content of the current vendor (s[i]). Taking the maximum of these two options ensures that m[i] stores the highest sugar consumption achieved so far.

By iterating through all the vendors and applying this expression, you can determine the maximum sugar you can gain from the street vendors and satisfy your sweet tooth.

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- Fixed cost: $4000, Marginal cost per item: $2.8, Price: $4

To identify the fixed cost, marginal cost per item, and the price at which the item is sold, we can analyze the given functions.

1. Fixed cost:
The fixed cost refers to the cost that remains constant regardless of the quantity produced or sold. In this case, the fixed cost is represented by the constant term in the total cost function. Looking at the equation C = 4000 + 2.8q, we can see that the fixed cost is $4000.

2. Marginal cost per item:
The marginal cost per item represents the additional cost incurred when producing or selling one more item. To find the marginal cost per item, we need to calculate the derivative of the total cost function with respect to the quantity (q).

Differentiating the total cost function C = 4000 + 2.8q with respect to q, we get:
dC/dq = 2.8

Therefore, the marginal cost per item is $2.8.

3. Price:
The price at which the item is sold is represented by the revenue per item. Looking at the revenue function R = 4q, we can see that the price at which the item is sold is $4.

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Therefore,[tex]f(x) = 7/x^5[/tex] and g(x) = x - 10 are two nontrivial functions that satisfy the given equation [tex]f(g(x)) = 7/(x - 10)^5[/tex].

Let's find the correct functions f(x) and g(x) such that [tex]f(g(x)) = 7/(x - 10)^5[/tex].

Let's start by breaking down the expression [tex]7/(x - 10)^5[/tex]. We can rewrite it as[tex](7 * (x - 10)^(-5)).[/tex]

Now, we need to find functions f(x) and g(x) such that f(g(x)) equals the above expression. To do this, we can try to match the inner function g(x) first.

Let's set g(x) = x - 10. Now, when we substitute g(x) into f(x), we should get the desired expression.

Substituting g(x) into f(x), we have f(g(x)) = f(x - 10).

To match [tex]f(g(x)) = (7 * (x - 10)^(-5))[/tex], we can set [tex]f(x) = 7/x^5[/tex].

Therefore, the functions [tex]f(x) = 7/x^5[/tex] and g(x) = x - 10 satisfy the equation [tex]f(g(x)) = 7/(x - 10)^5.[/tex]

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Answer:

For two triangles to be similar, their corresponding angles must be equal. Triangle 1 has angles measuring 26°, 53°, and an unknown angle. Triangle 2 has angles measuring 26°, a°, and an unknown angle.

To determine the value of a that would refute Frank's claim, we need to find a value for which the unknown angles in both triangles are equal.

In triangle 1, the sum of the angles is 180°, so the third angle can be found by subtracting the sum of the known angles from 180°:

Third angle of triangle 1 = 180° - (26° + 53°) = 180° - 79° = 101°.

For triangle 2 to be similar to triangle 1, the unknown angle in triangle 2 must be equal to 101°. Therefore, the value of a that would refuse Frank's claim is a = 101°.

Step-by-step explanation:

Answer:

101

Step-by-step explanation:

In Δ1, let the third angle be x

⇒ x + 26 + 53 = 180

⇒ x = 180 - 26 - 53

⇒ x = 101°

∴ the angles in Δ1 are 26°, 53° and 101°

In Δ2, if the angle a = 101° then the third angle will be :

180 - 101 - 26 = 53°

∴ the angles in Δ2 are 26°, 53° and 101°, the same as Δ1

So, if a = 101° then the triangles will be similar