A race car is running practice laps in preparation for an upcoming race. To judge how the car is performing, the crew takes measurements of the car's speed S(t) (in miles per hour, or mph) every minute. The measurements are given in the table below.

Question

t (min) --- S(t) (mph
0 --- 201
1 --- 205
2 --- 208
3 --- 214
4 --- 218
5 --- 212
6 --- 219
7 --- 223
8 --- 220
9 --- 221
10 --- 217
11 --- 218
12 --- 216

A. Use the trapezoid rule with 4 equal subdivisions to approximate the total distance the car traveled (in miles) over the first 12 minutes.

B. Find one approximation for S''(6), including the units. Explain what this quantity means in the context of the problem.

C. What was the car's average speed in mph over the first 12 minutes? If the car needs to have an average speed of 210 mph to qualify for the race, is it currently fast enough to qualify?

Answer 1

A. The total distance the car traveled over the first 12 minutes can be approximated using the trapezoid rule with 4 equal subdivisions as follows:

d = (1/2)(201 + 205 + 208 + 214 + 218 + 212 + 219 + 223 + 220 + 221 + 217 + 218 + 216) * (12/4) = 2,722 miles

B. One approximation for S''(6) is 4.5 mph/min. This quantity represents the rate of change of the car's acceleration at t = 6 minutes.

C. The car's average speed over the first 12 minutes is 214.5 mph. Since this is greater than 210 mph, the car is currently fast enough to qualify for the race.