Problem 43 - Calculus Early Transcendentals

Question

A large building shaped like a box is 50 \(\mathrm{m}\) high with a face that is \(80 \mathrm{m}\) wide. A strong wind blows directly at the face of the building, exerting a pressure of \(150 \mathrm{N} / \mathrm{m}^{2}\) at the ground and increasing with height according to \(P(y)=150+2 y,\) where \(y\) is the height above the ground. Calculate the total force on the building, which is a measure of the resistance that must be included in the design of the building.

Step-by-Step Solution

Answer

Answer: The total force exerted by the wind on the building is 800,000 N.

Step 1: Define the formula for pressure

The pressure increases with height according to the given formula: P(y) = 150 + 2y

Step 2: Calculate the force exerted by the wind at each height

The force exerted by the wind at a specific height, dF, can be represented as the pressure at that height multiplied by the area it's acting on. The area at each height is the width of the building multiplied by a small change in height, dy: dF = P(y) * 80 * dy

Step 3: Integrate the force equation with respect to the height to find the total force

To calculate the total force exerted by the wind on the building, integrate the force equation with respect to y from 0 to 50 m: F = ∫(150 + 2y) * 80 * dy from 0 to 50

Step 4: Solve the integral

Let's solve the integral: F = 80 * ∫(150 + 2y) dy from 0 to 50 F = 80 * (150y + y^2) evaluated from 0 to 50 Now, plug in the limits: F = 80 * [(150 * 50 + 50^2) - (150 * 0 + 0^2)] F = 80 * (7500 + 2500) F = 80 * 10000

Step 5: Calculate the total force

Finally, multiply the result by 80 to obtain the total force acted on the building's face: F = 80 * 10000 = 800000 N The total force acting on the building is 800,000 N. This is the resistance that must be included in the design of the building to withstand the force exerted by the strong wind.