Explanation:
It is given that, the force needed to keep a car from skidding on a curve varies inversely as the radius of the curve and jointly as the weight of the car and the square of the car's speed such that,
![F\propto \dfrac{mgv^2}{r}](https://tex.z-dn.net/?f=F%5Cpropto%20%5Cdfrac%7Bmgv%5E2%7D%7Br%7D)
![F=\dfrac{kmgv^2}{r}](https://tex.z-dn.net/?f=F%3D%5Cdfrac%7Bkmgv%5E2%7D%7Br%7D)
mg is the weight of the car
r is the radius of the curve
v is the speed of the car
Case 1.
F = 640 pounds
Weight of the car, W = mg = 2600 pound
Radius of the curve, r = 650 ft
Speed of the car, v = 40 mph
![640=\dfrac{k(2600)(40)^2}{650}](https://tex.z-dn.net/?f=640%3D%5Cdfrac%7Bk%282600%29%2840%29%5E2%7D%7B650%7D)
k = 0.1
Case 2.
Radius of the curve, r = 750 ft
Speed of the car, v = 30 mph
![F=\dfrac{0.1\times 2600\times (30)^2}{750}](https://tex.z-dn.net/?f=F%3D%5Cdfrac%7B0.1%5Ctimes%202600%5Ctimes%20%2830%29%5E2%7D%7B750%7D)
F = 312 N
Hence, this is the required solution.