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. Suppose that 200 pounds of force keeps a 3200-pound car from skidding on a curve of radius 400 ft at 25 mph. What force would keep the same car from skidding on a curve of radius 550 ft at 55 mph?
Let F represent the force needed to keep the car from skidding.
According to the given statement:
F = k * (W * v^2) / r
where:
F is the force needed to keep the car from skidding,
k is a constant of variation,
W is the weight of the car,
v is the speed of the car,
r is the radius of the curve.
From the initial conditions with a curve radius of 400 ft and a speed of 25 mph:
200 = k * (3200 * 25^2) / 400
Solving for k:
200 = k * (3200 * 625) / 400
200 = k * 2000000 / 400
200 = k * 5000
k = 200 / 5000
k = 0.04
Now we can apply the value of k to the new conditions with a curve radius of 550 ft and a speed of 55 mph:
F = 0.04 * (3200 * 55^2) / 550
Calculating:
F = 0.04 * (3200 * 3025) / 550
F = 0.04 * 9696000 / 550
F = 387840 / 550
F = 705.1636
Therefore, the force needed to keep the same car from skidding on a curve of radius 550 ft at 55 mph is approximately 705.16 pounds.