Find all solutions of the equation in radians.
sin(2t)cos(t)-cos(2t)sin(t)=0

The number of items that will maximize the profit is 2000. Thus, the correct answer is option c.

To calculate the maximum profit that can be earned we have to differentiate the equation and find the value of x

dP/dx = 1/dx (-3x² + 12x + 75)

= -6x + 12

Calculating dP/dx = 0

0 = -6x + 12

6x = 12

x = 2

Next, we calculate the next differential of the equation:

It comes out to be -6

Since it is smaller than zero, the value of x calculated is the maxima.

The maxima = 2

Thus, the item that will maximize the profit comes out to be 2000 as x is the number of items made in thousand.

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To translate one-step equations, you need to understand the language of algebra.

Algebraic expressions involve variables, numbers, and operations such as addition, subtraction, multiplication, and division. One-step equations require only one operation to isolate the variable, making them easy to solve.

To write an equation to represent the statement "15 is 9 more than j," you can use the equation 15 = j + 9. This equation says that 15 is equal to j plus 9. To solve for j, you need to isolate j on one side of the equation by subtracting 9 from both sides. This gives you the equation j = 6.

To solve the equation j % 3 = c, you need to understand the modulus operator, which gives you the remainder when two numbers are divided. In this case, j % 3 means the remainder when j is divided by 3. To solve for j, you need to multiply both sides of the equation by 3, which gives you the equation j = 3c.

In summary, to translate one-step equations, you need to understand the language of algebra and the operations involved. To solve for variables, you need to isolate them on one side of the equation. And to solve equations involving the modulus operator, you need to understand how it works and how to apply it to solve for variables.

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Answer:

The smaller rectangle has perimeter

2(4 + 6) = 2(10) = 20, so the larger rectangle will have perimeter 30. The dimensions of the larger rectangle are 6 by 9 since 2(6 + 9) = 2(15) = 30.

Answer: I believe the answer would be 38/5x - 3

The number of unique rhombuses that can be constructed using this information is three.

When given a 40° angle and an 8 cm line segment, we can construct three distinct rhombuses. A rhombus is a quadrilateral with all sides of equal length, and opposite angles are congruent.

In this scenario, the given 40° angle determines the orientation of the rhombus, while the 8 cm line segment determines its side length. By connecting the endpoints of the line segment with congruent opposite angles, we can create three different rhombuses.

Each rhombus formed will possess an angle measure of 40° and a side length of 8 cm. However, these rhombuses will vary in terms of their overall shape and orientation. Each one represents a unique configuration that satisfies the given angle and side length criteria.

Therefore, the correct answer is that three distinct rhombuses can be constructed using the given information.

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Step-by-step explanation:

(a) [1/2 x 1/4] + [1/2 x 6]

First, we can simplify each term separately:

1/2 x 1/4 = 1/8

1/2 x 6 = 3

Now, we can add these two simplified terms:

1/8 + 3 = 3 1/8

Therefore, [1/2 x 1/4] + [1/2 x 6] simplifies to 3 1/8.

(b) [1/5 x 2/15] - [1/5 x 2/15]

Both terms are the same, so when we subtract them, the result will be zero:

[1/5 x 2/15] - [1/5 x 2/15] = 0

Therefore, [1/5 x 2/15] - [1/5 x 2/15] simplifies to 0.

b(x) = 6x² and m(x) = -1/3x²- 4 have similarities and differences as quadratic functions with different coefficients and signs.

We can look at the similarities and differences to compare b(x) = 6x² and m(x) = -1/3x² - 4

Both b(x) and m(x) are quadratic functions, which means they have an x² term.

They both have a constant term, with b(x) having a constant of 0 and m(x) having a constant of -4.

The coefficients of the x² term are different: b(x) has a coefficient of 6, while m(x) has a coefficient of -1/3.

The signs of the coefficients are different: b(x) has a positive coefficient, while m(x) has a negative coefficient.

Overall, b(x) and m(x) have some similarities in their form as quadratic functions, but their coefficients and signs are different, which means they will have different shapes and behaviors.

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The formula for continuously compounded interest is:

A = Pe^(rt)

Where A is the ending amount, P is the principal, e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate as a decimal, and t is the time in years.

If we want to find how long it takes for the investment to double, we need to solve for t when A = 2P:

2P = Pe^(rt)

Dividing both sides by P and simplifying, we get:

2 = e^(rt)

Taking the natural logarithm of both sides, we get:

ln(2) = rt ln(e)

ln(2) = rt

t = ln(2) / r

Substituting the given values, we get:

t = ln(2) / 0.0425

t ≈ 16.3 years

So it will take approximately 16.3 years for the investment to double. Rounded to the nearest tenth of a year, the answer is 16.3 years.

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To take a picture we must press the shutter button pointing the lens towards the image we want to capture.

To take a picture we must follow the following steps. In general, we must have a camera at hand and know how to use it. There is a great diversity of cameras with different characteristics, but the basics to take a photo are the following:

In the first place, we must locate ourselves at a prudent distance from the element that we are going to photograph, making sure that it comes out completely in the camera's focus.

Once we have focused on the object, we must make sure that nothing is going to move the camera or go through between the camera and the object.

Later, we must make sure that there is enough light for the object to come out sharp in the photo.

Finally, we press the shutter and take the photo. In some cases we will have the digital photo or in others we will be able to print it on photographic paper.

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If the eastbound train travels at 75 miles per hour, it will take the two trains 2.8 hours to be 476 miles apart.

To solve the problem, we can use the formula:

distance = rate × time

Let's call the time it takes for the two trains to be 476 miles apart "t".

The westbound train travels at a rate of 95 miles per hour, so in time "t" it will travel a distance of 95t miles. Similarly, the eastbound train travels at a rate of 75 miles per hour, so in time "t" it will travel a distance of 75t miles.

To find the total distance between the two trains after time "t", we add the distances traveled by each train:

95t + 75t = 476

Combining like terms and solving for "t", we get:

170t = 476

t = 2.8 hours

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6 different permutations using the number 1 , 2 , 3 can be masde for the lock .

Given,

1 , 2 , 3 numbers to be used for a three digit lock .

There are 3 options for the first digit, 2 options for the second digit, and 1 option for the third digit.

To find the total number of permutations, we can use the formula for permutations:

Permutations of n items taken r at a time, which is n!/(n-r)!.

Here,

In this case,

n is 3

r is 3,

So the total number of permutations is 3!/(3-3)! = 3! = 3 x 2 x 1 = 6.

Hence,

So, you can make 6 different permutations using the numbers 1, 2 and 3 in any order.

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Answer:

Yes

Step-by-step explanation:

Yes, because its graph represents a straight line

Answer:

Step-by-step explanation:

1807

The difference in the proportions of firstborn or only children for the two populations from which these samples were drawn is 0.16.The bound for the error of estimation is 95% confidence that the true difference in proportions of firstborn or only children for the two populations is between 0.058 and 0.262.

To estimate the difference in the proportions of firstborn or only children for the two populations, we can use the sample proportions and apply the formula:

p1 - p2 = (x1/n1) - (x2/n2)

where p1 and p2 are the true population proportions, x1 and x2 are the numbers of firstborn or only children in the samples, and n1 and n2 are the sample sizes.

The sample proportions,

p1 = x1/n1 = 0.7

p2 = x2/n2 = 0.54

Substituting these values into the formula,

p1 - p2 = (x1/n1) - (x2/n2) = 0.7 - 0.54 = 0.16

Therefore, we estimate that the difference in the proportions of firstborn or only children for the two populations is 0.16.

To find a bound for the error of estimation, we can use the formula:

E = z sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)

where E is the margin of error, z is the critical value for the desired level of confidence (we'll use z = 1.96 for a 95% confidence interval), and p1 and p2 are the sample proportions.

Substituting the given values, we get:

E = 1.96sqrt(0.7(1-0.7)/180 + 0.54*(1-0.54)/100) ≈ 0.102

Therefore, we can say with 95% confidence that the true difference in proportions of firstborn or only children for the two populations is between 0.058 and 0.262

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The resulting concentration from the mixture will be: 4.67%

To obtain the concentration of the mixture, we will multiply the volumes of the substances by their percentages and then equate the result that we get to the sum of the mixture. This gives us:

100 g * 0.08 + 200 g * 0.03 = (100 + 200) C

8 + 6 = 300C

= 14 = 300 C

C = 14/300

= 0.046

This could be rewritten in the form of a percentage as 4.67%.

So, the resulting concentration of the solution will be 4.67%.

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The magnitude of the resultant force is approximately 119.89 pounds, and the direction angle is approximately S12.2°W.

To solve the problem, we can use vector addition.

Let F1 be the vector representing the first force, and F2 be the vector representing the second force. Then, we can find the resultant force R by adding the two vectors:

R = F1 + F2

To add two vectors, we need to resolve them into their x and y components. Let's do that first.

For F1:

Magnitude = 80 pounds

Direction = S58°E

To resolve F1 into its x and y components, we can use trigonometry:

Fx1 = 80 cos 58° = 42.57 pounds (east)

Fy1 = 80 sin 58° = 68.13 pounds (south)

For F2:

Magnitude = 50 pounds

Direction = N76°E

To resolve F2 into its x and y components, we can again use trigonometry:

Fx2 = 50 cos (180° - 76°) = -16.92 pounds (east)

Fy2 = 50 sin (180° - 76°) = 48.76 pounds (north)

Note that we used (180° - 76°) for the angle because the direction is N76°E, which means it is 76° east of due north.

Now we can add the x and y components separately:

Rx = Fx1 + Fx2 = 42.57 - 16.92 = 25.65 pounds (east)

Ry = Fy1 + Fy2 = 68.13 + 48.76 = 116.89 pounds (south)

To find the magnitude and direction of the resultant force, we can use trigonometry again:

Magnitude = sqrt(Rx^2 + Ry^2) = sqrt(25.65^2 + 116.89^2) = 119.89 pounds (rounded to the nearest hundredth)

Direction angle = atan(Rx/Ry) = atan(25.65/116.89) = 12.2° (rounded to the nearest tenth)

The direction angle is approximately S12.2°W, and the resultant force has a magnitude of about 119.89 pounds.

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The point on the curve y = -4x² - 2x - 11 where the tangent is parallel to the line 8x - 3 is (-1, -13).

To find the points on the curve where the tangent is parallel to the line, we need to find where the derivative of the curve is equal to the slope of the line.

The given curve is: y = -4x² - 2x - 11

The derivative of this curve is: y' = -8x - 2

The slope of the given line is: 8

We want to find the points where the derivative of the curve is equal to the slope of the line:

-8x - 2 = 8

Solving for x, we get:

x = -1

Now, we can plug this value of x back into the original equation to find the corresponding value of y:

y = -4(-1)² - 2(-1) - 11

y = -13

Therefore, the point on the curve where the tangent is parallel to the line 8x - 3 is (-1, -13).

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the cross product of the normal vectors of the planes will give you the direction vector of the line.

(5,−3,1)×(3,1,−5)=(14,28,14)

Which we can scale down to (1,2,1)

Now we need a point on the line. By inspection we can see that (1,1,0) lies in both planes.

Sometimes it it not that easy. But it is usually pretty easy to find a point in at least one plane and then travel along some line in that plane until we intersect the line in question.

Vector form of the line  L:(x,y,z)=(1,2,1)t+(1,1,0)

In parametric form  x=t+1,y=2t+1,z=t

The parametric equations are (x(t), y(t), z(t)) = (17/34 + 11t/34, 22/34 - 5t/34, 57/34 + 7t/34), where t is a parameter. The angle between the planes is 93.7 degrees.

To find the line of intersection of the planes, we can set the two equations equal to each other and solve for x, y, and z in terms of a parameter t. We can begin by eliminating one variable, say z.

From the first equation, we have z = 2 - 5x + 3y, and substituting this into the second equation gives 3x + y - 5(2 - 5x + 3y) = 4. Simplifying this equation, we get 22x - 14y - 23 = 0. Solving for y in terms of x, we get y = (22/14)x - (23/14).

Substituting this into the first equation and solving for z, we get z = (17/14)x + (57/14). Therefore, we have x = (17/22) + (11/22)t, y = (22/14) - (5/14)t, and z = (17/14)x + (57/14) + (7/22)t. These are the parametric equations for the line of intersection of the planes.

To find the angle between the planes, we can find the angle between their normal vectors.

The normal vector to the plane 5x - 3y + z = 2 is (5, -3, 1), and the normal vector to the plane 3x + y - 5z = 4 is (3, 1, -5). Using the dot product formula, we have cosθ = (5)(3) + (-3)(1) + (1)(-5) / sqrt(5² + (-3)² + 1²) sqrt(3² + 1² + (-5)²), which simplifies to cosθ = -19/34.

Taking the inverse cosine of this value, we get θ = 93.7 degrees, rounded to one decimal place. Therefore, the angle between the planes is approximately 93.7 degrees.

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The approximate area of the remaining pie is approximately 33.49 square inches.

To find the approximate area of the remaining pie, we need to subtract the area of the two pieces that were eaten from the total area of the pie.

The total area of the pie is given by the formula for the area of a circle:

[tex]Area = π * (radius)^2.[/tex]

Since the pie has a diameter of 8 inches, the radius is half of that, which is 4 inches. Plugging in the values:

[tex]Area = π * (4 inches)^2[/tex]

≈ 3.14 * 16 square inches

≈ 50.24 square inches.

Since the pie was cut into 6 equal slices, each slice represents 1/6th of the total area. So the area of the two pieces that were eaten is:

Area eaten = 2 * (1/6) * 50.24 square inches

≈ 16.75 square inches.

To find the area of the remaining pie, we subtract the area eaten from the total area:

Area remaining = Total area - Area eaten

= 50.24 square inches - 16.75 square inches

≈ 33.49 square inches.

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Answer: F, H, and J all have no real solution. The only equation that has a solution is

Step-by-step explanation: Use foil method.

The problem involves finding the derivative of a given function at a specified point.

Specifically, we are given the function f(x) = -8x^(-1) + 5x^(-2), and we need to find the value of the derivative f'(2) at x = 2. To find the derivative of f(x), we need to apply the rules of differentiation, which involve taking the derivative of each term separately and applying the power rule and chain rule as needed.

Once we have the derivative function f'(x), we can evaluate it at x = 2 to find the value of f'(2). Differentiation is a fundamental concept in calculus, and is used extensively in many areas of mathematics, science, and engineering. The ability to find derivatives allows us to analyze the behavior of functions and solve a wide variety of problems, from optimization to modeling physical systems.

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An object moves in simple harmonic motion with a period of 8 minutes and an amplitude of 12 m. At time =t0 minutes, its displacement d from rest is −12m, and initially, it moves in a positive direction. We can write the final equation for the displacement d as a function of time t: d(t) = 12 * cos((π/4)t + π)

To model the displacement d as a function of time t for an object in simple harmonic motion with a period of 8 minutes and an amplitude of 12m, we'll use the following equation:

d(t) = A * cos(ωt + φ)

where:
- d(t) is the displacement at time t
- A is the amplitude (12m in this case)
- ω is the angular frequency, calculated as (2π / period)
- t is the time in minutes
- φ is the phase angle, which we'll determine based on the initial conditions

Since the period is 8 minutes, we can calculate the angular frequency as follows:
ω = (2π / 8) = (π / 4)

At t = 0 minutes, the displacement is -12m, and the object moves in a positive direction. So we have:
-12 = 12 * cos(φ)

Dividing both sides by 12:
-1 = cos(φ)

Therefore, φ = π (or 180°) since the cosine of π is -1.

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1) Factorial - Numbers arranged in a specific order.

2) 0! - An array of numbers.

3) Sigma - A symbol for a sum.

4) Addition series - A series whose terms are formed by addition.

5) Combinations - 1.

6) Geometric series - A series whose terms are formed by multiplication.

7) ! - Permutation.

8) Sequence - A set of elements that does not consider order.

9) Pascal's triangle - A study of likelihoods.

10) Probability - A sum of numbers in a specific order.

11) Arithmetic series - A set of elements in a specific ord

1. Factorial: A product of all positive integers up to a given number (n!). For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

2. Array of numbers 0!: 0! is defined to be 1. This is a convention to simplify mathematical expressions and formulas.

3. Sigma (Σ): A symbol used to represent the sum of a series of numbers, typically written as Σ(expression) with specified lower and upper limits.

4. Series: A sequence of terms formed by adding the terms of a sequence. An example of an addition series is 1 + 2 + 3 + 4 + 5.

5. Combinations: The number of ways to choose a specific subset of items from a larger set without regard to the order in which they are chosen.

6. Geometric Series: A series whose terms are formed by multiplying each term by a constant factor. For example, 1, 2, 4, 8, 16 is a geometric series with a constant factor of 2.

7. Permutation: An arrangement of elements from a set where the order of the elements matters.

8. Sequence: A set of elements that does not consider order. It is a list of numbers or objects arranged according to a specific rule.

9. Pascal's Triangle: A triangular array of numbers in which the first and last number in each row is 1, and each of the other numbers is the sum of the two numbers above it. Pascal's Triangle is used to study the likelihoods and combinatorics.

10. Probability: A measure of the likelihood that a particular event will occur. It is the sum of the probabilities of all possible outcomes in a specific order, expressed as a number between 0 and 1.

11. Arithmetic Series: A set of elements in a specific order where each term is formed by adding a constant difference to the preceding term. For example, 2, 5, 8, 11, 14 is an arithmetic series with a constant difference of 3.

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The series 2n^2 is divergent.

To express the nth partial sum as a telescoping sum, we need to find a pattern in the terms of the series.

The general term of the series is given by an = 2n^2 - ?.

The nth partial sum can be written as:

sn = a1 + a2 + a3 + ... + an

= 2(1)^2 - ? + 2(2)^2 - ? + 2(3)^2 - ? + ... + 2n^2 - ?

We can simplify the above expression by factoring out 2 from each term:

sn = 2(1^2 + 2^2 + 3^2 + ... + n^2) - n?

Using the formula for the sum of squares, we have:

sn = 2(n(n+1)(2n+1)/6) - n?

Simplifying further, we get:

sn = (n^3 + 3n^2 + 2n)/3 - n?

Taking the limit as n approaches infinity, we get:

lim n->∞ sn = lim n->∞ [(n^3 + 3n^2 + 2n)/3 - n?]

Since the term n? grows without bound as n approaches infinity, the limit of sn does not exist.

Therefore, the series 2n^2 - ? is divergent.

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The absolute maximum value is 20/17, which occurs at x = 5, and the absolute minimum value is 4/5, which occurs at x = 1.

To find the absolute maximum and minimum values of the function f(x) = 8x/(6x + 4) on the interval (1, 5), we need to find the critical points of the function within the interval and evaluate the function at those points, as well as at the endpoints of the interval.

First, let's find the derivative of the function:

f(x) = 8x/(6x + 4)

f'(x) = [8(6x + 4) - 8x(6)] / (6x + 4)^2

f'(x) = [8(2)] / (6x + 4)^2

f'(x) = 16 / (6x + 4)^2

The critical points occur when f'(x) = 0 or is undefined. However, since f'(x) is always positive on the interval (1, 5), there are no critical points within the interval.

Next, let's evaluate the function at the endpoints of the interval:

f(1) = 8(1)/(6(1) + 4) = 8/10 = 4/5

f(5) = 8(5)/(6(5) + 4) = 40/34 = 20/17

Finally, we need to determine which of these values is the absolute maximum and which is the absolute minimum.

Since f(x) is always positive on the interval (1, 5), the function can never be less than 0. Therefore, the absolute minimum value is the smallest value of f(x) on the interval, which occurs at x = 5, where f(5) = 20/17.

To find the absolute maximum value, we compare the values of f(1), f(5), and the maximum value of f(x) as x approaches the endpoints of the interval. We can use the fact that the function is continuous on the closed interval [1, 5] to find the maximum value.

As x approaches 1, we have:

f(x) = 8x/(6x + 4) → 8/10 = 4/5

As x approaches 5, we have:

f(x) = 8x/(6x + 4) → 40/34 = 20/17

Therefore, the absolute maximum value is 20/17, which occurs at x = 5, and the absolute minimum value is 4/5, which occurs at x = 1.

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We are given that f is continuous on [1, 5], differentiable on (1, 5), and f'(x) ≤ 23 for all x. We want to find the largest possible value for f(5). the largest possible value for f(5) is found to be 96.

We can apply the Mean Value Theorem (MVT) here, which states that if a function is continuous on [a, b] and differentiable on (a, b), there exists a number c in the interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a). In this case, a = 1, b = 5, and f(1) = 4.

Since f'(x) ≤ 23 for all x, we know that f'(c) ≤ 23. Plugging into the MVT equation, we have:
[tex]f'(c) = (f(5) - f(1))/(5 - 1) ≤ 23, f'(c) = (f(5) - 4)/4 ≤ 23[/tex]

To find the largest possible value for f(5), we assume f'(c) is equal to its maximum, 23: 23 = (f(5) - 4)/4. Solving for f(5), we get: f(5) = 4 + 4 * 23 = 4 + 92 = 96. So, the largest possible value for f(5) is 96.

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Answer:

568.2 in

Step-by-step explanation:

To find this we first have to divide 1894/10 then we get 189.4 which we multiply by 3 to find how much more sand we need.

The budget with the highest savings amount that still meets Robert's basic needs is Budget C. The answer is C. Budget C

To determine which budget would help Robert most quickly achieve his financial goal of starting his own landscaping business, we need to compare the savings amounts in each budget.

Budget A:
Income: $1520
Expenses: Rent ($400), Utilities ($80), Food ($250), Cell Phone ($0), Savings ($400), Entertainment ($220), Clothing ($130)
Net Income: $40

Budget B:
Income: $1520
Expenses: Rent ($400), Utilities ($80), Food ($250), Cell Phone ($75), Savings ($600), Entertainment ($320), Clothing ($0)
Net Income: $20

Budget C:
Income: $1520
Expenses: Rent ($400), Utilities ($80), Food ($150), Cell Phone ($70), Savings ($500), Entertainment ($125), Clothing ($120)
Net Income: $75

Budget D:
Income: $1600
Expenses: Rent ($400), Utilities ($80), Food ($400), Cell Phone ($110), Savings ($260), Entertainment ($200), Clothing ($150)
Net Income: $0

In Budget C, Robert can save $500 per month while still covering his expenses for rent, utilities, food, cell phone, entertainment, and clothing. Additionally, this budget has a positive net income of $75, indicating that it is sustainable.

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The lateral surface area of the rectangular prism is given as follows:

L = 60 cm².

The lateral surface area of a rectangular prism of length l, width w and height h is given by the equation presented as follows:

L = 2 ( l + w ) h

The dimensions for this problem are given as follows:

l = 3 cm, w = 2 cm and h = 6 cm.

Hence the lateral surface area of the rectangular prism is given as follows:

L = 2 x (2 + 3) x 6

L = 60 cm².

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The speed of the van traveling south is 46 miles per hour.

Let the speed of the van traveling south be x miles per hour. Then, the speed of the van traveling north is (x + 10) miles per hour.

Since both vans are moving apart, we add their speeds: x + (x + 10) = 2x + 10 miles per hour.

In 2.5 hours, they are 255 miles apart. So, (2x + 10) * 2.5 = 255.

Now, we solve for x:

5x + 25 = 255
5x = 230
x = 46

The speed of the van traveling south is 46 miles per hour.

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