Stopping Distance not to exceed

Question

For a certain model of car the distance d required to stop the vehicle if it is traveling at v mi/h is given by the function d(v) = v + (v^2 / 25) where d is measured in feet. Kerry wants her stopping distance not to exceed 165 ft. At what range of speeds can she travel? (Assume v is positive. Round your values to one decimal place. Enter your answer using interval notation.)

Answer 1

Given,

Distance should not exceed 165 ft.

So d(v) <= 165,

d(v) = v + (v^2 /25) <= 165

=> 25v + v^2 <= 165 * 25

=> v^2 +25v - 4125 <=0

On solving above quadratic equation we get,

1/2 (-25 - 5 sqrt(685))<=v<=1/2 (5 sqrt(685) - 25)

or

-77.93 <= v <= 52.93

but since velocity is assumed to be non-negative.

Therefore,

0 <= v <= 52.93