Question

the lifting force, f, exerted on an airplane wing varies jointly as the area, a, of the wings surface and the square of the planes velocity, v. the lift of a wing with an area of 170 square feet is 15,900 pounds when the plane is going 90 miles per hour. find the lifting force on the wing if the plane speeds up to 130 miles per hour. (leave the variation constant in fraction form or round to at least 5 decimal places. round off your final answer to the nearest pound.)

Explanation:

Step1: Write the joint - variation formula

Since \(F\) varies jointly as \(A\) and \(v^{2}\), the formula is \(F = kAv^{2}\), where \(k\) is the variation constant.
We know that when \(A = 170\) and \(v=90\), \(F = 15900\). Substitute these values into the formula:
\(15900=k\times170\times90^{2}\)

Step2: Solve for \(k\)

First, calculate \(90^{2}=8100\). Then the equation becomes \(15900 = k\times170\times8100\).
\(k=\frac{15900}{170\times8100}=\frac{15900}{1377000}=\frac{159}{13770}\approx0.01154699\)

Step3: Find \(F\) when \(v = 130\) and \(A = 170\)

Substitute \(k\approx0.01154699\), \(A = 170\), and \(v = 130\) into \(F=kAv^{2}\).
\(v^{2}=130^{2}=16900\)
\(F=0.01154699\times170\times16900\)
\(F = 0.01154699\times2873000\)
\(F\approx3317\)

Answer:

\(3317\)