Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft², how much force will a wind of 80 mph place on a surface of 2 ft²?

검증된 단계별 안내

Identify the variables and the joint variation relationship. Let \(F\) be the force, \(A\) be the area, and \(v\) be the velocity. The problem states that \(F\) varies jointly as \(A\) and the square of \(v\), so we write the equation as: \[F = k \cdot A \cdot v^2\] where \(k\) is the constant of proportionality.

Use the given values to find the constant \(k\). Substitute \(F = 50\), \(A = \frac{1}{2}\), and \(v = 40\) into the equation: \[50 = k \cdot \frac{1}{2} \cdot (40)^2\]

Solve the equation for \(k\) by isolating it on one side: \[k = \frac{50}{\frac{1}{2} \cdot 40^2}\]

Use the value of \(k\) to find the force when \(v = 80\) mph and \(A = 2\) ft\(^2\). Substitute these values into the original formula: \[F = k \cdot 2 \cdot (80)^2\]

Calculate the force \(F\) by multiplying the constant \(k\) by the new area and the square of the new velocity to find the force exerted on the surface.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Joint Variation

Joint variation describes a relationship where a quantity varies directly as the product of two or more other quantities. In this problem, the force varies jointly as the area and the square of the velocity, meaning force = k × area × velocity², where k is a constant.

Solving for the Constant of Variation

To find the constant k in a joint variation problem, substitute the known values of the variables and the resulting value into the equation. This allows you to solve for k, which can then be used to find unknown values when other variables change.

Applying the Variation Formula to New Conditions

Once the constant k is known, you can calculate the force for different values of area and velocity by substituting them into the formula. This step involves careful substitution and arithmetic to find the new force exerted under changed conditions.

Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft2, how much force will a wind of 80 mph place on a surface of 2 ft2?