Question

How far will a car travel before coming to a full stop after the driver applies the brakes? How do accident investigators determine whether cars involved in accidents were traveling at safe speeds? They consider variables such as a car’s speed, the driver’s reaction time, the type of road, the weather conditions, and the effectiveness of the brakes.

Part A
Suppose a car is traveling on a dry, level road and the driver has average reaction time. The formula y = 0.048x2 + 0.37x relates the car’s approximate stopping distance y, in feet, to the speed of the car x, in miles per hour.

Choose an online tool or paper to graph the formula.


Part B
At about what speed should a driver travel to guarantee a stopping distance of 50 ft or less? Use the quadratic formula to find the speed. Round your answer to the nearest tenth.


 mi/h or less


Part C
Explain each step of your solution to Part B.


Part D
Explain how you can find the speed you found in Part B by using a graphing calculator to graph two functions.

Answer 1

Part A
Use an online graphing tool such as Desmos (www.desmos.com/calculator) or GeoGebra (www.geogebra.org). In Desmos type:
y = 0.048 x^2 + 0.37 x
Set the window to show x from 0 to about 60 and y from 0 to about 300 to see the curve clearly.

Part B
28.7 mi/h or less

Part C — step-by-step solution using the quadratic formula
1. Set the stopping-distance formula equal to 50 ft:
0.048 x^2 + 0.37 x = 50
2. Rearrange to standard quadratic form:
0.048 x^2 + 0.37 x − 50 = 0
Here a = 0.048, b = 0.37, c = −50.
3. Use the quadratic formula x = [−b ± sqrt(b^2 − 4ac)] / (2a).
Compute the discriminant:
b^2 − 4ac = (0.37)^2 − 4(0.048)(−50) = 0.1369 + 9.6 = 9.7369.
sqrt(9.7369) ≈ 3.1204.
4. Compute the two roots:
x = [−0.37 ± 3.1204] / (2·0.048) = [−0.37 ± 3.1204] / 0.096.
Positive root: x ≈ (−0.37 + 3.1204)/0.096 = 2.7504/0.096 ≈ 28.65.
Negative root: x ≈ (−0.37 − 3.1204)/0.096 ≈ −36.35 (not physically relevant).
5. Interpret: for the upward-opening parabola, y ≤ 50 between the two roots, so physically the allowable speeds are 0 up to the positive root. Rounding to the nearest tenth gives 28.7 mi/h. Thus to guarantee a stopping distance of 50 ft or less, drive at about 28.7 mi/h or slower.

Part D — using a graphing calculator
1. Enter the two functions: y1 = 0.048 x^2 + 0.37 x and y2 = 50.
2. Graph both curves. The horizontal line y = 50 will intersect the parabola at two points.
3. Use the calculator’s intersection feature to find the x-coordinates of the intersection points. The positive intersection is ≈ 28.65.
4. Conclude the same: speeds at or below ≈ 28.7 mi/h yield stopping distances ≤ 50 ft.