Respuesta :
Answer:
Approximately 7.04 feet
Step-by-step explanation:
Here's how to solve this using linear approximation:
Understanding Linear Approximation:
Linear approximation uses the tangent line to a function at a specific point to estimate the function's value near that point. The formula is:
f(x) ≈ f(a) + f'(a)(x-a)
Steps:
Define the function: You have the stopping distance function:
F(s) = 1.1s + 0.054s^2
Find the derivative: This represents the rate of change of stopping distance with respect to speed.
F'(s) = 1.1 + 0.108s
Set the point of approximation: You want to estimate at s = 55 mph.
Apply the formula:
F(55) = 1.1(55) + 0.054(55)^2 ≈ 223.55 ft (stopping distance at 55 mph)
F'(55) = 1.1 + 0.108(55) ≈ 7.04 (rate of change at 55 mph)
Now, for a 1 mph increase in speed:
F(56) ≈ F(55) + F'(55) (56 - 55)
F(56) ≈ 223.55 + 7.04 ≈ 230.59 ft
Calculate the change:
Change in stopping distance ≈ 230.59 - 223.55 = 7.04 ft
Conclusion:
Using linear approximation, we estimate that the stopping distance increases by approximately 7.04 feet for each additional mph when the speed is around 55 mph.
Important Note: Linear approximation is more accurate for small changes in speed. As the speed change increases, the accuracy might decrease.
