A population of rabbits on a farm grows by 12% each year. Define a sequence {rn} describing the rabbit population at the end of each year. Suppose that the sequence starts with r0 = 30.
(a) Give a mathematical expression for r12. (You don't have to actually compute the number.)
(b) If each rabbit consumes 10 pounds of rabbit food each year, then how much rabbit food is consumed in 10 years? For simplicity, you can omit the food consumed by the baby rabbits born in a given year. For example, suppose the farm starts tabulating rabbit food on January 1, 2012 at which time the rabbit population is 30. You will count the food consumed by those 30 rabbits during 2012. You won't count the food consumed by the rabbits born in 2012 until after January 1, 2013. Again, you don't have to compute the number, but you do have to give a closed form (without the summation) mathematical expression for the number.

The area of the region bounded by the curve r = 2sin(θ) in the sector 0≤θ≤π is π/2 square units.

The curve given by the polar equation r = 2sin(θ) is a sinusoidal spiral that starts at the origin, goes out to a maximum distance of 2 units, and then spirals back into the origin as θ increases from 0 to 2π. The sector 0≤θ≤π is half of this spiral, so we can find its area by integrating the area element dA = 1/2 r^2 dθ over this sector

A = ∫[0,π] 1/2 (2sin(θ))^2 dθ

Simplifying the integrand and applying the half-angle identity for sin^2(θ), we get

A = ∫[0,π] sin^2(θ) dθ

= ∫[0,π] (1 - cos^2(θ)) dθ

Integrating term by term, we get

A = [θ - 1/2 sin(2θ)]|[0,π]

= π/2 square units.

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cos(sin⁻¹ˣ-tan^-1y) can be written as: x/√(1+y²) + √(1-x²)/√(1+y²). This can be answered by the concept of Trigonometry.

We can use the trigonometric identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b) to write cos(sin⁻¹ˣ-tan^-1y) in terms of x and y.

Let a = sin⁻¹ˣ and b = tan^-1y, then we have:

cos(sin⁻¹ˣ-tan^-1y) = cos(a-b)

= cos(a)cos(b) + sin(a)sin(b)

= (√(1-x²))(1/√(1+y²)) + x/√(1+y²)

= x/√(1+y²) + √(1-x²)/√(1+y²)

Therefore, cos(sin⁻¹ˣ-tan^-1y) can be written as:

x/√(1+y²) + √(1-x²)/√(1+y²)

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a)  If the first person chosen is a boy, what is the probability that

the second person chosen is also a boy is: 12/27

b) The probability that both students chosen are girls is: 5/18

The parameters given are:

There are 28 students in a class

13 of the students are boys

According to the question we have

When first chosen a boy , then the rest is

28 - 1 = 27

Then the rest boys are 12

From 27, has 12 boys

The probability that the second person also is a boy = 12/27

b) There are:

28 - 13 = 15 girls

Probability that first is a girl = 15/28

Probability that second is a girl = 14/27

Thus:

P(both are girls) = (15/28) * (14/27) = 5/18

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Answer:

Step-by-step explanation:

use Pythagorean triangle:

a^{2} + b^{2} = c^{2}

a= 12

b= 16

c = ?

12^{2} + 16^{2} = c^{2}

144 + 256 = c^{2}

400 = c^{2}

\sqrt{400} = c

20 = c

c = 20 ft

Boundary value analysis is a testing technique used to identify defects or issues at the boundaries or limits of input values. True, in boundary value analysis both valid and invalid inputs are tested to verify potential issues.

Boundary value analysis is a testing technique used to identify defects or issues at the boundaries or limits of input values. The main idea is to test inputs that are just above, just below, and exactly at the specified boundaries or limits. This helps in uncovering potential issues that may arise due to boundary conditions.

Valid inputs are those that fall within the acceptable range of values, while invalid inputs are those that fall outside the acceptable range of values. Both valid and invalid inputs are tested during boundary value analysis to ensure thorough testing of the system under test. By testing valid inputs, we can verify if the system handles inputs within the acceptable range correctly. By testing invalid inputs, we can identify any issues or defects that may arise when inputs fall outside the acceptable range.

Therefore, in boundary value analysis, both valid and invalid inputs are tested to verify potential issues or defects in the system

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Answer:

Using a scale factor of -1/2, you can enlarge the center with the axis points, (-1,-1).

Step-by-step explanation:

In order to enlarge the triangle M, you would need to use the scale factor of -1/2.

With the center of enlargement then found on plotted axis (-1, -1), one would find a new triangle labeled N.

Answer: the quotient is 2x - 1 and the remainder is 0. So we can write:

(56x^2-60x+16) ÷ (28x-16) = 2x - 1.

Step-by-step explanation:

2x - 1

-------------

28x - 16 | 56x^2 - 60x + 16

56x^2 - 32x

--------------

-28x + 16

-28x + 16

----------

0

On quantities:

3 boxes.

10 devices.

28 phalange bones.

15 grams of fat.

2 smoothies left.

1256 oz of water.

2.5 miles.

7. The physician assistant sees an average of 37 x 3 = 111 patients over 3 days.

Since each patient requires 2 gloves, the total number of gloves needed is 111 x 2 = 222 gloves.

Since there are 100 gloves in each box, the number of boxes needed is 222/100 = 2.22, which rounds up to 3 boxes.

8. The medical sales rep sells 17 devices per day on average. To sell 500 devices in November, the sales rep needs to sell 500/30 = 16.67 devices per day on average.

The sales rep exceeds the goal by 17 - 16.67 = 0.33 devices per day on average.

Therefore, the sales rep exceeds the goal by 0.33 x 30 = 10 devices.

9. There are 56 - (14 x 2) = 28 phalange bones in each foot.

10. Frank consumed 41 grams of fat for lunch, so he has 56 - 41 = 15 grams of fat left to consume.

11. The rec center sells an average of 12 smoothies per day, so in 4 days, it will sell 12 x 4 = 48 smoothies.

Since there are 50 smoothies in each case, there will be 50 - 48 = 2 smoothies left in stock after 4 days.

12. Ashton drank 24 oz of water, so he needs to drink an additional 64 - 24 = 40 oz of water.

Since 1 oz = 0.03125 cups, Ashton needs to drink 40/0.03125 = 1280 cups of water.

Therefore, Ashton needs to drink 1280 - 24 = 1256 oz of water for the rest of the day.

13. Kathy walks 3 times a week for a total of 3 x 2.5 = 7.5 miles.

To meet her goal of 10 miles per week, Kathy needs to walk an additional 10 - 7.5 = 2.5 miles.

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Answer:

[tex]f^{-1}(x) = \dfrac{6x + 5}{1 - x}[/tex]

Step-by-step explanation:

To find the inverse of a function, we can swap x and y (f(x)), then solve for y, and represent that y as [tex]f^{-1}(x)[/tex].

[tex]f(x) = \dfrac{x - 5}{x + 6}[/tex]

↓ swapping x and y

[tex]x = \dfrac{y - 5}{y + 6}[/tex]

↓ multiplying both sides by (y + 6)

[tex]x(y + 6) = y - 5[/tex]

↓ simplifying using the distributive property

[tex]xy + 6x = y - 5[/tex]

↓ subtracting 6x and y from both sides to isolate the y terms

[tex]xy - y = - 6x - 5[/tex]

↓ undistributing y from the left side

[tex]y(x - 1) = - 6x - 5x[/tex]

↓ dividing both sides by (x - 1)

[tex]y = \dfrac{-6x - 5}{x-1}[/tex]

↓ (optional) multiplying the fraction by [tex]\bold{\dfrac{-1}{-1}}[/tex]

[tex]y = \dfrac{6x + 5}{1 - x}[/tex]

↓ replacing y with [tex]f^{-1}(x)[/tex]

[tex]\boxed{f^{-1}(x) = \dfrac{6x + 5}{1 - x}}[/tex]

The probability of getting a result as extreme or more extreme than 8 heads is 0.0547, which corresponds to answer choice (a).

The probability of getting a result as extreme or more extreme than 8 heads in one flip of 10 unbiased coins can be found using the binomial probability formula. We need to calculate the probability of getting exactly 8 heads, 9 heads, and 10 heads, then sum them up.

The binomial probability formula is: P(X=k) = C(n, k) × p^k × (1-p)^(n-k), where C(n, k) represents the number of combinations, n is the number of trials (in this case, 10 coin flips), k is the number of successful outcomes (heads), and p is the probability of success (0.5 for unbiased coins).

P(8 heads) = C(10, 8) × 0.5⁸ × 0.5² = 45 × 0.0039 × 0.25 = 0.0439
P(9 heads) = C(10, 9) × 0.5⁹ × 0.5¹ = 10 × 0.00195 × 0.5 = 0.0098
P(10 heads) = C(10, 10) × 0.5¹⁰ × 0.5⁰ = 1 × 0.00098 × 1 = 0.00098

Now, add these probabilities together: 0.0439 + 0.0098 + 0.00098 = 0.0547.

Therefore, the probability of getting a result as extreme or more extreme than 8 heads is 0.0547, which corresponds to answer choice (a).

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To find the surface area, you add up the area of the lateral faces and the area of the bases. The area of the triangular bases is 10.5 inches squared, and the area of the lateral faces is (3.5 * 9) + (4.5 * 9) + (3 * 9) = 99 inches squared. 10.5 + 99 = 109.5 inches squared

96

Explanation:

Write the prime factorization of both the numbers.

24=2×2×2×3

32=2×2×2×2×2

a. The probability of P(Z > 1.02) = 0.1539
b. P(Z ≤ -2.36) = 0.0091
c. P(0 ≤ Z ≤ 1.07) = 0.3577


1. To find the probabilities, you need to reference a standard normal (z) table.

2. For a. P(Z > 1.02), look up 1.02 on the z table. The corresponding value is 0.8461. Since the question asks for P(Z > 1.02), subtract the value from 1: 1 - 0.8461 = 0.1539.

3. For b. P(Z ≤ -2.36), look up -2.36 on the z table. The corresponding value is 0.0091. Since the question asks for P(Z ≤ -2.36), the value is already correct: 0.0091.

4. For c. P(0 ≤ Z ≤ 1.07), look up 1.07 on the z table. The corresponding value is 0.8577. Since the question asks for P(0 ≤ Z ≤ 1.07), subtract 0.5 (value for Z = 0): 0.8577 - 0.5 = 0.3577.

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First, let's recall that two matrices A and B are considered similar if there exists an invertible matrix P such that A = PBP⁻¹.

Now, let's use this definition to prove both parts of the question:
(a) We want to show that A − λI and B − λI are similar. To do this, we need to find an invertible matrix P such that (A − λI) = P(B − λI)P⁻¹.

Let's start by manipulating the equation A = PBP⁻¹ to get A − λI = P(B − λI)P⁻¹.
Now, let's substitute this into the equation we want to prove:
A − λI = P(B − λI)P⁻¹

We want to show that this is equivalent to:
A − λI = Q(B − λI)Q⁻¹
for some invertible matrix Q.

To do this, let's try to manipulate the equation we have into the form we want:

A − λI = P(B − λI)P⁻¹
A − λI = PBP⁻¹ − λP(P⁻¹)
A − λI = PBP⁻¹ − λI
A = PB(P⁻¹) + λI

Now, let's try to get this into the form we want:

A = Q(B − λI)Q⁻¹
A = QBQ⁻¹ − λQ(Q⁻¹)
A = QBQ⁻¹ − λI
A = QB(Q⁻¹) + λI

Comparing the two equations, we see that if we let Q = P, we get the equation we want:

A − λI = PBP⁻¹ − λI
A − λI = QBQ⁻¹ − λI
Thus, A − λI and B − λI are similar.

(b) We want to show that det(A − λI) = det(B − λI).
From part (a), we know that A − λI and B − λI are similar, so there exists an invertible matrix P such that A − λI = P(B − λI)P⁻¹.
Now, let's take the determinant of both sides:
det(A − λI) = det(P(B − λI)P⁻¹)
det(A − λI) = det(P)det(B − λI)det(P⁻¹)
det(A − λI) = det(B − λI)
since det(P) and det(P⁻¹) cancel out.

Therefore, det(A − λI) = det(B − λI).

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The global maximum value of f(x, y) on the closed triangular region occurs at either (4, 0) or (0, 4), both of which have a value of 16.

The global minimum value of f(x, y) occurs at the critical point (0, 0), with a value of 0

To find the Optimization of multivariable functions i.e, global extreme values of [tex]f(x, y) = x^2 - xy + y^2[/tex] on the closed triangular region in the first quadrant bounded by the lines x = 4, y = 0, and y = x,

We need to first find the critical points of the function in the interior of the region and evaluate the function at these points, and then evaluate the function at the boundary points of the region.

To find the critical points of the function in the interior of the region, we need to solve the system of partial derivatives:

[tex]df/dx = 2x - y = 0\\f/dy = -x + 2y = 0[/tex]

Solving this system of equations, we get the critical point (x, y) = (0, 0).

To check whether this point is a maximum or a minimum, we need to evaluate the second partial derivatives of f:

[tex]d^2f/dx^2 = 2\\d^2f/dy^2 = 2\\d^2f/dxdy = -1[/tex]

The determinant of the Hessian matrix is:

[tex]d^2f/dx^2 \times d^2f/dy^2 - (d^2f/dxdy)^2 = 4 - 1 = 3[/tex]

Since this determinant is positive and [tex]d^2f/dx^2 = d^2f/dy^2 = 2[/tex] are both positive, the critical point (0, 0) is a local minimum.

Next, we need to evaluate the function at the boundary points of the region. These are:

(4, 0): f(4, 0) = 16

(0, 0): f(0, 0) = 0

(0, 4): f(0, 4) = 16

(y, y) for 0 ≤ y ≤ 4: [tex]f(y, y) = 2y^2 - y^2 = y^2[/tex]

Therefore, the global maximum value of f(x, y) on the closed triangular region occurs at either (4, 0) or (0, 4), both of which have a value of 16.

The global minimum value of f(x, y) occurs at the critical point (0, 0), with a value of 0.

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Answer:

Step-by-step explanation:

Using a standard normal table, we can find the area under the curve between 1.20 and 1.85 to be approximately 0.4678. Therefore, the answer is (c) 0.4678.

a) The general solution is y(x) = y_c(x) + y_p(x) = c1e^x + c2e^(4x) + 8ex.

b) The general solution is y(x) = y_c(x) + y_p(x) = c1e^(x/2)cos((√3/2)x) + c2e^(x/2)sin((√3/2)x) - (1/4)sin(3x).

For the differential equation y'' - 5y' + 4y = 8ex, the characteristic equation is r^2 - 5r + 4 = 0, which has roots r1 = 1 and r2 = 4. Thus, the complementary function is y_c(x) = c1e^x + c2e^(4x).

To find the particular solution, we guess a solution of the form y_p(x) = Ae^x. Then, y_p''(x) - 5y_p'(x) + 4y_p(x) = Ae^x - 5Ae^x + 4Ae^x = Ae^x. We need this to equal 8ex, so we set A = 8, and the particular solution is y_p(x) = 8ex.

Thus, the general solution is y(x) = y_c(x) + y_p(x) = c1e^x + c2e^(4x) + 8ex.

b) For the differential equation y'' - y' + y = 2sin(3x), the characteristic equation is r^2 - r + 1 = 0, which has roots r1,2 = (1 ± i√3)/2. Thus, the complementary function is y_c(x) = c1e^(x/2)cos((√3/2)x) + c2e^(x/2)sin((√3/2)x).

To find the particular solution, we guess a solution of the form y_p(x) = A sin(3x) + B cos(3x). Then, y_p''(x) - y_p'(x) + y_p(x) = -9A sin(3x) - 9B cos(3x) - 3A cos(3x) + 3B sin(3x) + A sin(3x) + B cos(3x) = -8A sin(3x) - 6B cos(3x). We need this to equal 2sin(3x), so we set A = -1/4 and B = 0, and the particular solution is y_p(x) = (-1/4)sin(3x).

Thus, the general solution is y(x) = y_c(x) + y_p(x) = c1e^(x/2)cos((√3/2)x) + c2e^(x/2)sin((√3/2)x) - (1/4)sin(3x).

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Different sequence of answers is  59049.

Explanation: -  

To determine the number of different sequences of answers that can be created with ten multiple-choice questions, each having three possible answers, we need to use the multiplication principle of counting. This principle states that the total number of possible outcomes of a sequence of events is the product of the number of outcomes for each event.

For the first question, there are three possible answers. For the second question, there are three possible answers, and so on for each of the ten questions. Using the multiplication principle, we can determine the total number of different sequences of answers by multiplying the number of outcomes for each question together: 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 59,049

Therefore, there are 59,049 different sequences of answers that can be created with ten multiple-choice questions, each having three possible answers.

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The equation to compute the area A of the circle is: [tex]A = π(r^2 - r0^2) + A0[/tex] where r0 is the initial radius and A0 is the initial area.

The equation to compute the area A of a circle with radius r is [tex]A = πr^2[/tex].

Using this equation and the given information that the radius is increasing at a rate of centimeters per second, we can write:

[tex]\frac{dA}dt} = 2rπ \frac{dr}{dt}[/tex]

where dA/dt represents the rate of change of area with respect to time, and [tex]\frac{dr}{dt}[/tex] represents the rate of change of radius with respect to time.

Part 1:

If we want to find the area of the circle at a specific time t, we can integrate both sides of the equation with respect to time:

[tex]\int\limits dA= \int\limits 2πr \frac{dr}{dt}  \, dt[/tex]

Integrating both sides gives:

[tex]A = πr^2 + C[/tex]

where C is the constant of integration. Since we are given the initial radius, we can use it to find the value of C:

When t = 0, r = r0

[tex]A = πr0^2 + C[/tex]

Therefore, [tex]C = A - πr0^2[/tex]

Substituting this value of C back into the equation gives:

[tex]A = πr^2 + A - πr0^2[/tex]

Simplifying gives:

[tex]A =π(r^2 - r0^2) + A0[/tex]

where A0 is the initial area of the circle.

Therefore, the equation to compute the area A of the circle is:

[tex]A = π(r^2 - r0^2) + A0[/tex]

where r0 is the initial radius and A0 is the initial area.

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The general solution of differential equation is, y = k * (x²-9).

We can begin by separating the variables of the differential equation:

y′ = (2y) / (x²-9)

y′ / y = 2 / (x²-9)

Now we can integrate both sides with respect to their respective variables:

[tex]\int \dfrac{y'}{y} dy = \int \dfrac{2}{x^2-9} dx[/tex]

ln|y| = ln|x²-9| + C

where C is the constant of integration.

Simplifying:

|y| = e^(ln|x²-9|+C) = e^C * |x²-9|

Since e^C is a positive constant, we can write:

y = k * (x²-9)

where k is a non-zero constant. Therefore, the general solution of the given differential equation is y = k(x²-9), where k is any non-zero constant.

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--The complete question is, Find the general solution of the given differential equation. y′ = (2y) / (x²-9).--

The surface area of the composite figure is 1,665.04 in².

The volume of composite figure is 1,079.66 in³.

The volume  and surface area of the composite figure is calculated by applying the following formula as shown below;

The surface area = area of cone + area of hemisphere

S.A = πr(r + l) + 3πr²

S.A = π x 10 (10 + 13)  +  3π(10²)

S.A = 1,665.04 in²

The volume of composite figure is calculated as follows;

V = ¹/₃πr²h  +  ²/₃πr²

The height of the cone is calculated;

h = √(13² - 10²)

h = 8.31 in

V = ¹/₃π(10)²(8.31)  +  ²/₃π(10)²

V = 870.22 + 209.44

V = 1,079.66 in³

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The probability of at least one cloudy day in a five-day span is 0.7627 or approximately 0.76.

The probability of a sunny day in Virginia in July is 0.75, which means the probability of a cloudy day is 1 - 0.75 = 0.25.

Assuming the days are independent, the probability of at least one cloudy day in a five-day span can be calculated using the complement rule:

P(at least one cloudy day) = 1 - P(no cloudy days)

The probability of no cloudy days in a five-day span is the probability that all five days are sunny, which is [tex](0.75)^5[/tex] = 0.2373.

Therefore, the probability of at least one cloudy day in a five-day span is:

P(at least one cloudy day) = 1 - P(no cloudy days) = 1 - 0.2373 = 0.7627

So the probability of at least one cloudy day in a five-day span is 0.7627 or approximately 0.76.

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The minimum number of supports required is 7.

To determine the minimum number of supports required for the twenty-four feet (six 4-ft sections) of track lighting to be installed in a continuous row in a retail store, follow these steps:

1. Determine the total length of the track lighting: 6 sections * 4 feet per section = 24 feet.

2. Consider that a support is needed at the beginning and end of the track.

3. Assess the spacing between supports. For instance, let's assume supports can be placed every 4 feet.

4. Calculate the number of supports in between the ends: (24 feet - 4 feet) / 4 feet = 5 supports.

5. Add the supports at the beginning and end: 5 supports + 2 supports = 7 supports.

The minimum number of supports required is 7.

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If the inputs of a J-K flip-flop are J= 1 and K = 1 while the outputs are Q = 0 and Q= 1, the outputs after the next clock pulse occurs are C) Q=1, Q=0. An eight-line multiplexer must have D) eight data inputs and three select inputs.

For the first question, with the J-K flip-flop:
Given inputs J = 1 and K = 1, and outputs Q = 0 and Q' = 1. After the next clock pulse occurs, the outputs will be:
A) Q = 0, Q' = 0
B) Q = 1, Q' = 1
C) Q = 1, Q' = 0
D) Q = 0, Q' = 1
Answer: Since the J-K flip-flop is in toggle mode when J = 1 and K = 1, the outputs will toggle. Therefore, the correct answer is C) Q = 1, Q' = 0.
For the second question, regarding an eight-line multiplexer:
A) four data inputs and three select inputs.
B) eight data inputs and two select inputs.
C) eight data inputs and four select inputs.
D) eight data inputs and three select inputs.
Answer: An eight-line multiplexer requires three select inputs to choose from eight data inputs ([tex]2^3[/tex] = 8). Therefore, the correct answer is D) eight data inputs and three select inputs.

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The probability that fewer than 2 students come to office hours on any given Monday is 0.70.

To find the probability that fewer than 2 students come to office hours on any given Monday, we need to calculate the sum of the probabilities of X=0 and X=1.

P(X < 2) = P(X = 0) + P(X = 1)

= 0.40 + 0.30

= 0.70

From the given probability distribution, we can see that the probability of X=0 is 0.40 and the probability of X=1 is 0.30. These represent the probabilities of no students or one student showing up for office hours, respectively.

To find the probability that fewer than 2 students come to office hours on any given Monday, we need to add these probabilities together since X can only take on integer values.

Therefore, P(X < 2) = P(X = 0) + P(X = 1) = 0.40 + 0.30 = 0.70.

This means that there is a 70% chance that either no students or one student will show up for office hours on any given Monday.

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It would take Deshaun 5 hours to drive 335 miles at the same rate.

The SI unit of speed is m/s, and speed is defined as the ratio of distance to time. It is the shift in an object's location with regard to time.

We can use the formula:

rate = distance / time

to solve the problem. The rate is constant, so we can use it to find the time for a different distance.

First, we find Deshaun's rate:

rate = distance / time = 469 miles / 7 hours = 67 miles per hour

Now we can use this rate to find the time it would take to drive 335 miles:

time = distance / rate = 335 miles / 67 miles per hour

time = 5 hours

Therefore, it would take Deshaun 5 hours to drive 335 miles at the same rate.

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The complete question is:

Deshaun drove 469 miles in 7 hours. At the same rate, how long would it take him to drive 335 miles?

The volume of the region that bounds e is 15/2.

To visualize the region bounded by the parabolic cylinder, planes, and the plane z = 4, we can plot the surfaces using a 3D graphing software or by hand.

The parabolic cylinder z - 1/2 y^2 is a cylinder that opens upwards along the z-axis and its cross-sections perpendicular to the z-axis are parabolas. The planes x = 0 and z = 0 bound the cylinder on the left and at the bottom, respectively. The plane x = 1 bounds the cylinder on the right, and the plane z = 4 bounds it from above.

The intersection of the parabolic cylinder and the plane z = 4 is a parabolic curve in the plane z = 4. The intersection of the parabolic cylinder and the plane x = 1 is a straight line segment that runs along the y-axis from y = -2 to y = 2. The intersection of the parabolic cylinder and the plane z = 0 is the x-y plane, which contains the bottom of the cylinder.

To find the region that bounds e, we need to find the points where the parabolic cylinder intersects the planes x = 0, x = 1, and z = 1, and then determine the region that lies between these curves.

The intersection of the parabolic cylinder and the plane x = 0 is the y-axis. Therefore, the left boundary of the region is y = -2 and the right boundary is y = 2.

The intersection of the parabolic cylinder and the plane x = 1 is a line segment along the y-axis from y = -2 to y = 2. Therefore, the region is bounded on the left by the y-axis and on the right by the line segment x = 1, y = z^2/2 + 1/2.

The intersection of the parabolic cylinder and the plane z = 1 is a parabolic curve in the plane z = 1. To find the equation of this curve, we substitute z = 1 into the equation of the parabolic cylinder:

1 - 1/2 y^2 = x

Solving for y^2, we get:

y^2 = 2 - 2x

Therefore, the equation of the parabolic curve in the plane z = 1 is:

y = ±sqrt(2 - 2x)

The region bounded by the parabolic cylinder, planes, and the plane z = 4 is

therefore the region is given by:

0 ≤ x ≤ 1
-y/2 + 1/2 ≤ z ≤ 4
-y ≤ x^2/2 - 1/2

To visualize this region in 3D, we can plot the parabolic cylinder and the planes x = 0, x = 1, and z = 1 and shade the region between them. Then, we can extend this region upwards to the plane z = 4 to obtain the full region that bounds e.

To find the volume of this region, we can integrate the function 1 over this region with respect to x, y, and z:

∫∫∫_R 1 dV

where R is the region defined by the inequalities above. However, this triple integral is difficult to evaluate directly, so we can use the fact that the region is symmetric about the y-axis to simplify the integral by integrating first with respect to y and then with respect to x and z:

V = 2∫∫∫_R 1 dV

where the factor of 2 accounts for the symmetry of the region. Integrating with respect to y first, we get:

V = 2∫_{-2}^{2} ∫_{y^2/2 - 1/2}^{1/2} ∫_{-y/2 + 1/2}^{4} 1 dz dx dy

Evaluating this integral, we get:

V = 15/2

Therefore, the volume of the region that bounds e is 15/2.

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The value of c is -10/3. This can be answered by the concept of Differentiation.

To find the values of c for which the parabolas y = 9x² and x = c + 3y² intersect at right angles, we need to consider the slopes of the tangent lines at the intersection points.

First, let's find the derivatives of both functions to get the slopes:

For y = 9x², let's find dy/dx:
dy/dx = 18x

For x = c + 3y², let's find dx/dy:
dx/dy = 1 / (6y)

At the intersection points, we have:
9x² = y
c + 3y² = x

Since the tangent lines are perpendicular, their slopes multiply to -1:
(18x)(1 / (6y)) = -1

Now, substitute y = 9x² into the equation:
(18x)(1 / (6 × 9x²)) = -1
(18x)(1 / (54x²)) = -1
(1 / (3x)) = -1

Solving for x, we get x = -1/3.

Now substitute this value of x into the equation for y:
y = 9(-1/3)²
y = 9(1/9)
y = 1

So the intersection point is (-1/3, 1). Now substitute the value of y back into the equation for x to find c:
-1/3 = c + 3(1²)
-1/3 = c + 3
c = -1/3 - 3
c = -10/3

Therefore, the value of c is -10/3.

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