The table below shows the data for a car stopping on a wet road. What is the approximate stopping distance for a car traveling 35 mph?
Car Stopping Distances
| v (mph) | d |
|---|---|
| 15 | 17.9 |
| 20 | 31.8 |
| 50 | 198.7 |
[tex]d(v)=\frac{215 v^2}{644 f}[/tex]
A. 41.7 ft
B. 49.7 ft
C. 97.4 ft
Answer (2)
Estimate the constant f using the given data points and the formula.
Calculate the average value of f from the estimates.
Substitute v = 35 mph and the average f into the formula to find the stopping distance.
The approximate stopping distance for a car traveling at 35 mph is 97.4 ft .
Explanation
Understanding the Problem We are given a formula for the stopping distance d of a car traveling at speed v on a wet road: d ( v ) = 644 f 215 v 2 , where f is a constant. We are also given a table of data points for v and d . Our goal is to find the approximate stopping distance for a car traveling at 35 mph.
Estimating the Value of f First, we need to estimate the value of f using the given data. We can use any of the data points to solve for f . Let's use the data points (15, 17.9), (20, 31.8), and (50, 198.7) to find three estimates for f .
Calculating f from the first data point Using the first data point (15, 17.9), we have: 17.9 = 644 f 215 ( 1 5 2 ) . Solving for f , we get: f = 644 ( 17.9 ) 215 ( 1 5 2 ) = 644 × 17.9 215 × 225 = 11527.6 48375 ≈ 4.196
Calculating f from the second data point Using the second data point (20, 31.8), we have: 31.8 = 644 f 215 ( 2 0 2 ) . Solving for f , we get: f = 644 ( 31.8 ) 215 ( 2 0 2 ) = 644 × 31.8 215 × 400 = 20480 86000 ≈ 4.199
Calculating f from the third data point Using the third data point (50, 198.7), we have: 198.7 = 644 f 215 ( 5 0 2 ) . Solving for f , we get: f = 644 ( 198.7 ) 215 ( 5 0 2 ) = 644 × 198.7 215 × 2500 = 127910.8 537500 ≈ 4.200
Calculating the average f Now, we calculate the average value of f : f a vg = 3 4.196 + 4.199 + 4.200 ≈ 4.198757516543555 ≈ 4.20 .
Calculating the stopping distance Now that we have an estimate for f , we can plug in v = 35 into the formula to find the stopping distance: d ( 35 ) = 644 f a vg 215 ( 3 5 2 ) = 644 × 4.199 215 ( 1225 ) = 2704.156 263375 ≈ 97.402 .
Final Answer The approximate stopping distance for a car traveling at 35 mph is approximately 97.4 ft.
Examples
Understanding stopping distances is crucial for road safety. For instance, knowing that a car traveling at 35 mph requires approximately 97.4 feet to stop on a wet road helps drivers maintain a safe following distance. This concept extends to urban planning, where traffic light timings and intersection designs must account for these distances to prevent accidents. Moreover, in autonomous vehicle development, accurate stopping distance calculations are essential for ensuring safe navigation and collision avoidance, highlighting the practical importance of this mathematical relationship in everyday life.
To find the stopping distance for a car traveling at 35 mph, we estimated the constant f using provided data points and the formula. We calculated an average f and substituted it into the formula to find that the approximate stopping distance is 97.4 feet. Therefore, the answer is C. 97.4 ft.
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