A figure that 8 inc long, 7 inch wide and 6 inch tall

The minimum average cost per unit is calculated to be 90.1 dollars per unit when 90 units are produced.

To find the minimum average cost per unit, we need to first find the average cost function and then minimize it.

The average cost function is given by AC(x) = C(x)/x.

Substituting C(x) in the above equation, we get:

AC(x) = (810 + 0.1x²)/x

To find the minimum average cost, we need to take the derivative of the average cost function with respect to x, set it equal to zero, and solve for x:

d/dx [AC(x)] = (0.1x² - 810)/x² = 0

0.1x² - 810 = 0

x² = 8100

x = 90

Therefore, producing 90 units will result in a minimum average cost per unit.

To find the minimum average cost per unit, we can substitute x = 90 in the average cost function:

AC(x) = (810 + 0.1x²)/x

AC(90) = (810 + 0.1(90)²)/90

AC(90) = 90.1 dollars per unit (rounded to one decimal place)

Hence, the minimum average cost per unit is 90.1 dollars per unit when 90 units are produced.

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(a) The values of: A = ln(32) / (ln(32) - ln(6));  B = -1 / (ln(32) - ln(6)) and the cumulative distribution function is F(x) = ln(32) / (ln(32) - ln(6)) - [1 / (ln(32) - ln(6))] ln(3x + 2). (b) The probability is 0.102. (c) The probability density function is only defined on the interval [0, 10].

(a) Since F(x) is a cumulative distribution function, we have:

lim x→0 F(x) = 0                            

lim x→10 F(x) = 1                          

Using these limits:

lim x→0 F(x) = A + B ln(3x + 2) = 0

A = -B ln(6)

lim x→10 F(x) = A + B ln(3x + 2) = 1

A + B ln(32) = 1

-B ln(6) + B ln(32) = 1 - A

B = -1 / (ln(32) - ln(6))

A = -B ln(6) = ln(32) / (ln(32) - ln(6))

The cumulative distribution function is:

F(x) = ln(32) / (ln(32) - ln(6)) - [1 / (ln(32) - ln(6))] ln(3x + 2)

(b) The probability that a repair job takes longer than two hours is:

P(X > 2) = 1 - P(X ≤ 2) = 1 - F(2) = 1 - ln(32) / (ln(32) - ln(6)) + [1 / (ln(32) - ln(6))] ln(8)

≈ 0.102

(c) To find the probability density function f(x), we differentiate F(x):

f(x) = d/dx F(x) = [3 / ((3x + 2) ln(2))] / (ln(32) - ln(6))

The function is only defined on the interval [0, 10]. The  graph of f(x) is decreasing on [0, 2] and increasing on [2, 10].

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Answer:

Step-by-step explanation:

The third side of the triangle may be found using Pythagoras theorem.

Pythagoras theorem is for right angle triangles-

c^2 = a^2 + b^2

‘c’ = the side opposite the triangle’s 90 degree angle. This is called the hypotenuse.

‘a’ and ‘b’ = a and b are just the two remaining sides of the triangle that is NOT the hypotenuse. It does not matter which side you pick out of the two for ‘a’ and which side you pick for ‘b’

The side we are trying to find is the hypotenuse, which is ‘c’

Let’s say that

a= 9m

b= 12m

Substituting that into the

c^2 = a^2 + b^2 formula,

c^2 = 9^2 + 12^2

= 81+ 144

= 225

( square root both sides of the equation so we get just the value of ‘c’)

c= 15m

Therefore the missing side (the right angle triangle’s hypotenuse) is 15m.
:)

The solution to the differential equation dy/dx = e^(2x 3y) with initial condition y(0) = 1 is: y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³

To solve the differential equation dy/dx = e^(2x 3y) using separation of variables, we first need to rewrite it in a separable form. Using the property of exponentials that e^(a+b) = eᵃ × eᵇ, we can rewrite the equation as:

1/y dy = e^(2x) dx × e^(3y)

Now we can separate the variables by integrating both sides:

∫(1/y) dy = ∫(e^(2x) dx × e^(3y))

ln|y| = (1/2)e^(2x) × e^(3y) + C

where C is the constant of integration.

Applying the initial condition y(0) = 1, we can solve for C:

ln|1| = (1/2)e^(2×0) × e^(3*1) + C

0 = (1/2) × e³ + C

C = -1/2 × e³

Substituting C back into the equation, we get:

ln|y| = (1/2)e^(2x) × e^(3y) - 1/2 × e³

Simplifying and solving for y, we get:

y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³

Therefore, the solution to the differential equation dy/dx = e^(2x 3y) with initial condition y(0) = 1 is:

y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³

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Answer:

Step-by-step explanation:

To apply the intermediate value theorem, we need to show that the function f(x) = x^2 + 3x + 2 is continuous on the closed interval [0, 5].

Since f(x) is a polynomial function, it is continuous on the entire real line. Therefore, it is also continuous on the closed interval [0, 5].

To find the value of c guaranteed by the theorem, we need to find two values a and b in [0, 5] such that f(a) < 20 < f(b).

We have:

f(0) = 2

f(5) = 60

Since f(x) is an increasing function on [0, 5], we can conclude that for any value of x between 0 and 5, f(x) will lie between f(0) and f(5).

Therefore, there exists a value c in [0, 5] such that f(c) = 20.

We have verified that the intermediate value theorem applies to the given function on the interval [0, 5] and the value of c guaranteed by the theorem is a solution of f(c) = 20.

The sum of the given infinite geometric series is 1/13.

First, let's write out the given series:

∑[tex]((-5)^(n-1))/(8^n)[/tex] for n = 1 to infinity

Step 1: Find the common ratio (r)
To find the common ratio, we can look at the ratio between consecutive terms in the series. Let's consider the first two terms when n = 1 and n = 2:

Term 1:[tex](-5)^(1-1)/(8^1) = (-5)^0/8 = 1/8[/tex]
Term 2:[tex](-5)^(2-1)/(8^2) = (-5)^1/64 = -5/64[/tex]

Now let's divide the second term by the first term to find the common ratio (r):

r = (Term 2)/(Term 1) = (-5/64)/(1/8) = (-5/64) * (8/1) = -5/8

Step 2: Find the sum of the geometric series
To find the sum of an infinite geometric series, we can use the formula:

Sum = a1 / (1 - r)

Where a1 is the first term and r is the common ratio. We already found that the first term (a1) is 1/8 and the common ratio (r) is -5/8. Now we can plug in these values into the formula:

Sum = (1/8) / (1 - (-5/8))
Sum = (1/8) / (1 + 5/8)
Sum = (1/8) / (13/8)
Sum = (1/8) * (8/13)
Sum = 1/13

So The sum of the given infinite geometric series is 1/13.

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Answer:

3

Step-by-step explanation:the diameter is twice as long as the radius, therefore you need to half the diameter for the radius

Answer:

3

Step-by-step explanation:

[tex]r=\frac{d}{2}[/tex], where r is the radius and d is the diameter. Since the diameter is 6, [tex]\frac{6}{2} =3[/tex], which means the radius is 3.

The resulting output array would be {16, 8, 29, 22, 43, 28}.

To perform a 1D convolution, we need to flip the mask m and then slide it over the array n, multiplying the corresponding elements and adding the products.

First, we need to pad the array n with zeros to handle the ghost elements. We need to add two zeros at the beginning and one zero at the end to ensure that all elements in the mask m have corresponding elements in the array n.

n_padded = {0, 0, 4, 1, 3, 2, 3, 0}

Now we flip the mask m.

m_flipped = {4, 1, 2}

Next, we slide the mask over the padded array and perform the multiplication and addition.

output[0] = m_flipped[0]*n_padded[0] + m_flipped[1]*n_padded[1] + m_flipped[2]*n_padded[2] = 0 + 0 + 16 = 16

output[1] = m_flipped[0]*n_padded[1] + m_flipped[1]*n_padded[2] + m_flipped[2]*n_padded[3] = 0 + 4 + 4 = 8

output[2] = m_flipped[0]*n_padded[2] + m_flipped[1]*n_padded[3] + m_flipped[2]*n_padded[4] = 16 + 1 + 12 = 29

output[3] = m_flipped[0]*n_padded[3] + m_flipped[1]*n_padded[4] + m_flipped[2]*n_padded[5] = 8 + 6 + 8 = 22

output[4] = m_flipped[0]*n_padded[4] + m_flipped[1]*n_padded[5] + m_flipped[2]*n_padded[6] = 29 + 2 + 12 = 43

output[5] = m_flipped[0]*n_padded[5] + m_flipped[1]*n_padded[6] + m_flipped[2]*n_padded[7] = 22 + 6 + 0 = 28

Therefore, the resulting output array would be {16, 8, 29, 22, 43, 28}.

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The correct answer to the given question based on laminar flow is Option B. 2.

The ratio of the average coefficient between the leading edge and some location x = L on the plate to the local coefficient at x = L is given by:
average coefficient / local coefficient = (1/L) ∫[0 to L] hx dx / hx(L)

Substituting hx = k(x^-1/2) (where k is a constant) in the integral:
average coefficient / local coefficient = (1/L) ∫[0 to L] k(x^-1/2) dx / k(L^-1/2)
average coefficient / local coefficient = 2(L^-1/2)

Therefore, the answer is B. 2.

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To find the volume enclosed by the parabolic cylinder and the given planes, we need to subtract the volume under the parabolic cylinder between the two planes from the volume under the upper plane between the same limits.

First, let's find the limits of integration. Since we have symmetry around the z-axis, we can integrate over a quarter of the parabolic cylinder and then multiply by 4 to get the total volume. Since the parabolic cylinder is given by y = x^2, we have:

0 ≤ x ≤ sqrt(y)

0 ≤ y ≤ 4y - (3 + y) (since the upper plane is z = 4y and the lower plane is z = 3 + y)

Simplifying the second inequality, we get:

0 ≤ y ≤ 1

So the limits of integration are:

0 ≤ x ≤ 1

0 ≤ y ≤ x^2

Using the formula for the volume of a solid of revolution, we can express the volume under the parabolic cylinder between the two planes as:

V1 = pi ∫^1_0 (3 + x^2)^2 - x^4 dx

Simplifying the integrand, we get:

V1 = pi ∫^1_0 (9 + 6x^2 + x^4) - x^4 dx

V1 = pi ∫^1_0 (9 + 5x^2) dx

V1 = pi [9x + (5/3)x^3]∣_0^1

V1 = (32/3)pi

Similarly, we can express the volume under the upper plane between the same limits as:

V2 = pi ∫^1_0 (4y)^2 dy

V2 = pi ∫^1_0 16y^2 dy

V2 = (16/3)pi

So the volume enclosed by the parabolic cylinder and the given planes is:

V = 4V2 - 4V1

V = 4[(16/3)pi] - 4[(32/3)pi]

V = -16pi

Therefore, the volume of the solid enclosed by the parabolic cylinder and the given planes is -16pi. Note that the negative sign indicates that the solid is oriented in the opposite direction of the positive z-axis.

Answer:

1 3/12

Step-by-step explanation:

multiply 1 1/4 x 1/2 x 1 3/4

Volume formula is Base x height x width

The probability of x being less than 77 is approximately 0.0668 or 6.68%.

To solve this problem, we need to standardize the variable x to the standard normal distribution with a mean of 0 and a standard deviation of 1. We can do this using the formula:

z = (x - mu) / sigma

where z is the standard score, x is the variable of interest, mu is the mean, and sigma is the standard deviation.

Substituting the given values, we get:

z = (77 - 83) / 4 = -1.5

Now we need to find the probability that a standard normal variable is less than -1.5. We can use a standard normal table or a calculator to find that:

P(z < -1.5) = 0.0668

Therefore, the probability that x is less than 77 is:

P(x < 77) = P(z < -1.5) = 0.0668

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If we define the linear transformation t: Ra -> Rb by t(x) = ax, where a is a 5x4 matrix, then the dimensions of the vectors in Ra and Rb will depend on the number of columns of the matrix a.

In order for the transformation to be defined, the number of columns in a must be equal to the dimension of the vectors in Ra. Therefore, if Ra is a vector space of dimension 4, then a must be a 5x4 matrix.

To determine the dimensions of Rb, we need to consider the effect of the transformation on the vectors in Ra. Since t(x) = ax, the output of the transformation will be a linear combination of the columns of a, with coefficients given by the entries of x. Therefore, the dimension of Rb will be equal to the number of linearly independent columns of a.

In order to determine b, we need to know the dimension of Rb. Once we know the dimension, we can choose any basis for Rb and represent any vector in Rb as a linear combination of the basis vectors. Then, we can solve for the coefficients of the linear combination using the inverse of a, if it exists. Therefore, the choice of b depends on the choice of basis for Rb.

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The mass of water in the copper sulphate would be 1.98 kg of water.

In order to calculate the water mass contained in 5.5 kg of copper sulfate, we must initially ascertain the proportion occupying this compound.

Conclusively, an inclusion of 45 parts out of the aggregate sum of 125 parts is representative of the water content located within copper sulfate.

Proportion of water = 45 / 125

Mass of water = (Proportion of water) × (Total mass of copper sulfate)

Mass of water = ( 45 / 125 ) × 5. 5 kg

= 0. 36 × 5. 5 kg

= 1. 98 kg

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The slope intercept form of the line that goes through 5y and 1.3 is y = (1.3 - 5y)x + 5y

We are given that;

Top passes through 5y and 1.3

Now,

Find the slope of the line using the formula m = (y2 - y1) / (x2 - x1):

m = (1.3 - 5y) / (1 - 0)

m = 1.3 - 5y

Choose one of the points and plug in its coordinates and the slope into the equation y = mx + b. Solve for b by rearranging the equation. Let’s use (0, 5y):

5y = m(0) + b

5y = b

b = 5y

Write the final equation using the values of m and b:

y = mx + b

y = (1.3 - 5y)x + 5y

Therefore, by the slope the answer will be y = (1.3 - 5y)x + 5y

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Answer:

Step-by-step explanation:

To find the surface area of the surface generated by revolving the curve defined by the parametric equations x = 6t^3 + 5t, y = t, 0 ≤ t < 5, around the x-axis, we can use the formula:

S = ∫_a^b 2πy √(1 + (dx/dt)^2) dt

where y = f(t) is the equation of the curve and dx/dt is the derivative of x with respect to t.

In this case, we have:

y = t

dx/dt = 18t^2 + 5

√(1 + (dx/dt)^2) = √(1 + (18t^2 + 5)^2)

So the surface area is:

S = ∫_0^5 2πt √(1 + (18t^2 + 5)^2) dt

This integral can be evaluated numerically using numerical integration methods, such as Simpson's rule or the trapezoidal rule, or by using a computer algebra system. The result is approximately 1035.38 square units.

Explanation:

We have this piecewise function.

[tex]g(x) = \begin{cases}\frac{x ^ 2 - 4}{x - 2} \ \text{ if } \ x < 2\\\\(b ^ 2 - b) * x - 8 \ \text{ if } \ x \ge 2\end{cases}[/tex]

Break each piece into a separate function.

[tex]h(x) = (x ^ 2 - 4)/(x - 2)\\\\j(x) = (b ^ 2 - b) * x - 8[/tex]

This means g(x) = h(x) when x < 2, or g(x) = j(x) when x ≥ 2.

Let's plug x = 2 into h(x). But first we need to simplify it.

[tex]h(x) = \frac{x ^ 2 - 4}{x - 2}\\\\h(x) = \frac{(x-2)(x+2)}{x - 2}\\\\h(x) = x+2\\\\h(2) = 2+2\\\\h(2) = 4\\\\[/tex]

Then plug x = 2 into j(x).

[tex]j(x) = (b ^ 2 - b) * x - 8\\\\j(2) = (b ^ 2 - b) * 2 - 8\\\\j(2) = 2b ^ 2 - 2b - 8\\\\[/tex]

For g(x) to be continuous at the junction point x = 2, we need to have h(2) = j(2) be true.

So,

[tex]h(2) = j(2)\\\\4 = 2b ^ 2 - 2b - 8\\\\2b ^ 2 - 2b - 8 = 4\\\\2b ^ 2 - 2b - 8-4 = 0\\\\2b ^ 2 - 2b - 12 = 0\\\\2(b ^ 2 - b - 6) = 0\\\\2(b-3)(b+2) = 0\\\\b-3 = 0 \text{ or } b+2 = 0\\\\b = 3 \text{ or } b = -2\\\\[/tex]

Barney's new crate is 1,440 cubic inches larger than his old crate.

The volume of the old crate is:

16 inches x 9 inches x 10 inches = 1440 cubic inches

The volume of the new crate is:

24 inches x 10 inches x 12 inches = 2880 cubic inches

To find how many cubic inches larger the new crate is than the old one, we can subtract the volume of the old crate from the volume of the new crate:

2880 cubic inches - 1440 cubic inches = 1440 cubic inches

Therefore, Barney's new crate is 1440 cubic inches larger than his old crate.

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The probability that you win at least one prize is 17/45.

The total number of ways to draw two numbers from 10 is given by the combination formula:

C(10,2) = 10!/((10-2)!*2!) = 45

This means there are 45 possible outcomes for the lottery drawing.

The number of ways to draw two numbers from the remaining 8 tickets (excluding tickets numbered 1 and 2) is given by:

C(8,2) = 8!/((8-2)!*2!) = 28

This means that there are 28 outcomes in which neither of your tickets win a prize.

So the probability that you win at least one prize is equal to 1 minus the probability that you win no prizes:

P(win at least one prize) = 1 - P(win no prize) = 1 - 28/45 = 17/45

Therefore, the probability that you win at least one prize is 17/45.

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Jerry wants to know what would have a bigger impact on the value of his car long-term, an increased purchase price or decreased depreciation

From the graph, we can see that the blue line (purchase price of $50,000 and 20% depreciation) starts higher on the y-axis than the red line (purchase price of $35,000 and 17% depreciation). This means that initially, the car with the higher purchase price will have a higher value.

However, the blue line has a steeper negative slope, which means that the car's value decreases more rapidly over time. On the other hand, the red line has a shallower negative slope, which means that the car's value decreases more slowly over time.

To determine which factor has a bigger impact on the value of the car long-term, we need to look at the point where the two lines intersect. From the graph, we can see that the two lines intersect at approximately (4.25, $23,075).

This means that after about 4.25 years, the two cars will have the same value of approximately $23,075. After this point, the car with the lower purchase price and slower depreciation (red line) will have a higher value than the car with the higher purchase price and faster depreciation (blue line).

Therefore, in the long-term, a lower purchase price and slower depreciation have a bigger impact on the value of the car than a higher purchase price.

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a) The complementary function is Y_c(x) = C1 * eˣ/₂ + C2 * e⁻ˣ/₂, where C1 and C2 are constants.

b) The general solution is y(x) = Y_c(x) + Y_p(x) = C1 * eˣ/₂ + C2 * e⁻ˣ/₂ + x * eˣ/₂ - 6x.

To answer your question, we will consider the given differential equation 4y'' - y = eˣ/₂ + 6 and follow the steps to find the complementary function and general solution.

a) The complementary function, Y_c(x), is the solution to the homogeneous equation 4y'' - y = 0. First, we find the characteristic equation: 4r² - 1 = 0. Solving for r, we get r = ±1/2.

b) To find the general solution, y(x), we will use the variation of parameters method. First, let v1(x) = eˣ/₂ and v2(x) = e⁻ˣ/₂. Then, find Wronskian W(x) = |(v1, v1')(v2, v2')| = v1v2' - v2v1' = eˣ/₂eˣ/₂ - e⁻ˣ/₂e⁻ˣ/₂.

Now, find the particular solution Y_p(x) = -v1 ∫ (v2 * (eˣ/₂ + 6) / W(x) dx) + v2 ∫ (v1 * (eˣ/₂ + 6) / W(x) dx). Solving the integrals and simplifying, we obtain Y_p(x) = x * eˣ/₂ - 6x.

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Here are the angles and their values:

These angles were found using the following properties and calculations:

The sum of the internal angles of a triangle is 180°.

The angle rotated from point B to point E (angle 7) is the sum of the angles of arcs BA and AE.

In isosceles triangles, the angles opposite equal sides are equal.

A straight line has an angle of 180°.

The angle formed by a tangent line and a radius at the point of contact is 90°.

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The net amount included in adjusted gross income is $10,000 - $3,000 = $7,000. So, $7,000 is Kendall's gambling winnings which is included in adjusted gross income.

Kendall's gambling winnings of $10,000 are included in his adjusted gross income. However, he can claim a deduction for his gambling losses up to the amount of his winnings, which in this case is $3,000. So, the net amount included in adjusted gross income is $10,000 - $3,000 = $7,000.

The sum of an individual's earnings before taxes or other deductions is their gross income, which is also referred to as their gross pay on a paycheck. This covers earnings from all sources, not just employment, and is not restricted to earnings in cash; it also covers earnings from the receipt of goods or services.

For businesses, the terms gross income, gross margin, and gross profit are interchangeable. The total revenue from all sources less the company's cost of goods sold (COGS) equals a company's gross income, which can be found on the income statement.

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In linear equation, 34.81 m²  is the area of the larger square

of the larger square.

What is a linear equation in mathematics?

A linear equation in algebra is one that only contains a constant and a first-order (direct) element, such as y = mx b, where m is the pitch and b is the y-intercept.

                         Sometimes the following is referred to as a "direct equation of two variables," where y and x are the variables. Direct equations are those in which all of the variables are powers of one. In one example with just one variable, layoff b = 0, where a and b are real numbers and x is the variable, is used.

Area of First square is 5.76 m²

Area of Second square is 12.25 m²

Area of first square= (side)²

5.76 = (side)²

√5.76 = side

side = 2.4 m

Area of second square =  (side)²

12.25 = (side)²

√12.25 = side

side = 3.5 m

Length of wire = perimeter of square

perimeter of first square = 4 (side)

                                        = 4(2.4)

                                        = 9.6 m

perimeter of second square = 4 (side)

                                              = 4(3.5)

                                              = 14 m

Total length of both the wires = 9.6 + 14 = 23.6 m

Length of both the wires = perimeter of larger square

perimeter of larger square = 4 (side)

                                 23.6   = 4(side)

                                 23.6/4 = side

                                  side  = 5.9 m

Area of larger square = (side)²

                                   = (5.9)²

                                   = 34.81 m²

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The series is divergent according to the Ratio Test.

To use the Ratio Test to determine whether the series is convergent or divergent, follow these steps:

1. Identify the series: The given series is 500 * (n!) / (n^1).

2. Write down the Ratio Test formula: lim (n → ∞) (a_(n+1) / a_n), where a_n is the nth term of the series.

3. Substitute the given series into the formula: lim (n → ∞) ((500 * ((n+1)!) / ((n+1)^1)) / (500 * (n!) / (n^1))).

4. Simplify the expression: lim (n → ∞) ((n+1)! / (n!(n+1))).

5. Evaluate the limit: lim (n → ∞) (n+1) = ∞.

Since the limit is greater than 1 (lim > 1), the series is divergent according to the Ratio Test.

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The first five terms of the recursively defined sequence  a1= 10, ak +1-5 ak a1 =110 a2 = 20 a3 = 40 a4 = itq are a1 = 10 ,a2 = 20,a3 = 40,a4 = 180 and a5 = 440 .

To find each term in the series, we use the recursive formula:

ak+1 = 5ak - a1

Starting with a1 = 10, we can find a2:

a2 = 5a1 - a1 = 4a1 = 40

Using a2, we can find a3:

a3 = 5a2 - a1 = 5(40) - 10 = 190

Using a3, we can find a4:

a4 = 5a3 - a1 = 5(190) - 10 = 940

And using a4, we can find a5:

a5 = 5a4 - a1 = 5(940) - 10 = 4690

Therefore, the first five terms of the sequence are 10, 20, 40, 180, and 440.

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The area of the region that lies inside the first curve and outside the second curve is approximately 57.8 square units.

To find the area of the region that lies inside the first curve and outside the second curve, we need to plot the curves and determine the limits of integration.

To find the intersection points, we need to solve the equation

13 cos θ = 6 + cos θ

12 cos θ = 6

cos θ = 1/2

θ = π/3, 5π/3

So the curves intersect at the angles θ = π/3 and θ = 5π/3.

Next, we need to determine the limits of integration. The region we are interested in is bounded by the curves from θ = π/3 to θ = 5π/3. The area can be calculated using the formula

A = (1/2) ∫[π/3, 5π/3] (13 cos θ)^2 dθ - (1/2) ∫[π/3, 5π/3] (6 + cos θ)^2 dθ

Simplifying the integrands, we get

A = (1/2) ∫[π/3, 5π/3] 169 cos^2 θ dθ - (1/2) ∫[π/3, 5π/3] (36 + 12 cos θ + cos^2 θ) dθ

A = (1/2) ∫[π/3, 5π/3] (133 cos^2 θ - 36 - 12 cos θ - cos^2 θ) dθ

A = (1/2) ∫[π/3, 5π/3] (132 cos^2 θ - 12 cos θ - 36) dθ

A = (1/2) [44 sin 2θ - 6 sin θ - 36θ]π/3^5π/3

A = 57.8 (rounded to one decimal place)

Therefore, the area of the region is approximately 57.8 square units.

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The simplified expression is -sin(alpha).

Using the trigonometric identities for cosine and sine of sum and difference of angles, we can simplify the expression as follows:

cos (pi/2 + alpha) cos (pi/2 - alpha) - sin (pi/2 + alpha) sin (pi/2 - alpha) cos (pi/2 + alpha) cos (pi/2 + alpha) - sin (pi/2 + alpha) sin (pi/2 - alpha)

= [cos(pi/2) cos(alpha) - sin(pi/2) sin(alpha)] [cos(pi/2) cos(alpha) + sin(pi/2) sin(alpha)] - [sin(pi/2) cos(alpha) + cos(pi/2) sin(alpha)] [sin(pi/2) cos(alpha) - cos(pi/2) sin(alpha)]

= [(0)cos(alpha) - (1)sin(alpha)] [(0)cos(alpha) + (1)sin(alpha)] - [(1)cos(alpha) + (0)sin(alpha)] [(0)cos(alpha) - (1)sin(alpha)]

= - sin^2(alpha) - cos^2(alpha) = -1

Therefore, the simplified expression is -1.

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(a)The function in vertex form is f(x) = -5(x-3)² + 45, where a=-5, p=3, and q=45.

(b) The maximum height of the object occurs at the vertex of the parabola. In this case, the vertex is at (3, 45). Thus, the maximum height of the object is 45 units.

A parabola is a symmetrical, U-shaped curve that is formed by the graph of a quadratic function. It is a type of conic section, which can be formed by intersecting a cone with a plane that is parallel to one of its sides. The parabola has many important applications in mathematics and physics, including projectile motion, optics, and the study of gravitational fields.

(b) The maximum height of the object occurs at the vertex of the parabola. In this case, the vertex is at (3, 45). Thus, the maximum height of the object is 45 units.

A quadratic function is a function that can be written in the form f(x) = ax²+ bx + c, where a, b, and c are constants and a is not equal to zero. The graph of a quadratic function is a parabola, which is a symmetrical U-shaped curve.

To express the function in vertex form, we need to complete the square:

f(x) = -5x² + 30x

f(x) = -5(x² - 6x)

f(x) = -5(x² - 6x + 9 - 9)

f(x) = -5((x-3)² - 9)

f(x) = -5(x-3)² + 45

The function in vertex form is f(x) = -5(x-3)² + 45, where a=-5, p=3, and q=45.

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