Question

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?

Accepted Answer

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.

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?

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Key Concepts

Joint Variation

Solving for the Constant of Variation

Applying the Variation Formula to New Conditions