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?

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.

To solve the quadratic inequality, we first need to rearrange it in the standard form: ax^2 + bx + c ≤ 0.

Given: f(x) = 1/12 x^2 + 4/5x ≤ 200.

Multiply both sides by 12 to clear the rational expression:
12 * f(x) = x^2 + (48/5)x ≤ 2400.

Rearrange the inequality:
x^2 + (48/5)x - 2400 ≤ 0.

To solve this quadratic inequality, we can use either factoring, completing the square, or the quadratic formula. In this case, factoring might be convenient.

We factorize the quadratic equation:
(x - 40)(x + 60) ≤ 0.

Now, we can apply the sign test or make use of the fact that the product of two factors is negative when one factor is positive and the other is negative.

Considering the cases:
1. (x - 40) < 0 and (x + 60) > 0
2. (x - 40) > 0 and (x + 60) < 0

Case 1: (x - 40) < 0 and (x + 60) > 0
Solving (x - 40) < 0:
x < 40

Solving (x + 60) > 0:
x > -60

Combining x < 40 and x > -60, we get:
-60 < x < 40.

Case 2: (x - 40) > 0 and (x + 60) < 0
Solving (x - 40) > 0:
x > 40

Solving (x + 60) < 0:
x < -60

Combining x > 40 and x < -60, we get nothing feasible in this case.

Therefore, the valid solution is:
-60 < x < 40.

This means that the safe driving speed on the curve, where a driver can see the road ahead at most 200 ft, lies between -60 mph to 40 mph. However, we can consider only positive values for speed, so the safe speed limit for this curve would be between 0 mph and 40 mph.