The ceiling of a building has a height above the floor given by
z = 20 +
1
4
x,
and one of the walls follows a path modeled by y = x3/2. Find the surface area of the wall if 0 ≤ x ≤ 40. (All measurements are in feet. Round your answer to two decimal places.)
To find the surface area of the wall, we need to determine the length of the wall and the height of the wall.
First, let's find the length of the wall. The path of the wall is given by the equation y = x^(3/2). To find the total length of the wall between 0 ≤ x ≤ 40, we can calculate the definite integral of y with respect to x over the given range.
∫[0, 40] x^(3/2) dx
To integrate x^(3/2), we add 1 to the exponent and divide by the new exponent.
∫[0, 40] x^(3/2) dx = (2/5) * x^(5/2) ∣[0, 40]
Plugging in the limits of integration:
(2/5) * (40^(5/2)) - (2/5) * (0^(5/2))
= (2/5) * (40^(5/2))
≈ 1057.43
So, the length of the wall is approximately 1057.43 feet.
Next, let's find the height of the wall. The height of the wall is given by the equation z = 20 + (1/4)x. Since the wall follows the path of the ceiling, the height of the wall will be the same as the ceiling at every point.
The height of the wall is therefore 20 + (1/4)x.
Now we can find the surface area of the wall. The surface area can be calculated by multiplying the length of the wall by the height of the wall.
Surface Area of Wall = Length of Wall * Height of Wall
= 1057.43 * (20 + (1/4)x)
To find the total surface area of the wall between 0 ≤ x ≤ 40, we can integrate the expression for the surface area over the given range:
∫[0, 40] (1057.43 * (20 + (1/4)x)) dx
Integrating the expression:
= 1057.43 * (20x + (1/8)x^2) ∣[0, 40]
= 1057.43 * ((20 * 40) + (1/8) * 40^2) - 1057.43 * 0
= 1057.43 * (800 + (1/8) * 1600)
≈ 1057.43 * 1200
≈ 1268923.6
So, the surface area of the wall is approximately 1268923.6 square feet. Rounded to two decimal places, the answer is 1268923.60.