Highway engineers often use quadratic functions to model safe distance for cars. For example, f(x)= 1/2 x^2 + 4/5x is sometimes used to model the stopping distance for a car traveling at X miles per hour on dry, level pavement. If a driver can see only 200 ft ahead on a highway with a sharp curve, then safe driving speed X satisfy the quadratic inequality 1/12 x^2+ 4/5x ≤ 200. Solve this inequality to determine safe speed on a curve where a driver can see the road ahead at most 200 ft. What might be a safe speed limit for this curve?

Question

Highway engineers often use quadratic functions to model safe distance for cars. For example, f(x)= 1/2 x^2 + 4/5x is sometimes used to model the stopping distance for a car traveling at X miles per hour on dry, level pavement. If a driver can see only 200 ft ahead on a highway with a sharp curve, then safe driving speed X satisfy the quadratic inequality 1/12 x^2+ 4/5x ≤ 200. Solve this inequality to determine safe speed on a curve where a driver can see the road ahead at most 200 ft. What might be  a safe speed limit for this curve?

MathMate · Answer

Since it is not stated in the question, we assume that the unit of f(x) is in feet.

There is inconsistency with the two expressions for f(x), namely
f(x)=(1/2)x²+(4/5)x, and later
f(x)=(1/12)x²+(4/5)x

Solving the first. If the second expression is intended, you can solve similarly as below:

f(x)=(1/2)x²+(4/5)x ≤ 200
(1/2)x²+(4/5)x-200 ≤0
Using quadratic formula:
a=1/2
b=4/5
c=-200
x=-20.8 or +19.2 mi/hr
Reject negative root, so
x=19.2 mi/hr.

To stop within 200 feet, the speed must be ≤ 19.2 mi/hr.