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.
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?
A. To approximate the total distance traveled over the first 12 minutes using the trapezoid rule, we can divide the time into 4 equal subdivisions.
The average speed over each subdivision can be estimated by taking the average of the speeds at the beginning and end of the subdivision.
Subdivision 1: (201 + 205) / 2 = 203 mph
Subdivision 2: (205 + 208) / 2 = 206.5 mph
Subdivision 3: (208 + 214) / 2 = 211 mph
Subdivision 4: (214 + 218) / 2 = 216 mph
Now, we can use the trapezoid rule formula to calculate the approximate distance traveled in each subdivision:
Subdivision 1: (1/2) * (203) * (1-0) = 101.5 miles
Subdivision 2: (1/2) * (206.5) * (2-1) = 206.5 miles
Subdivision 3: (1/2) * (211) * (3-2) = 211 miles
Subdivision 4: (1/2) * (216) * (4-3) = 216 miles
To find the total distance traveled, we sum up the distances from each subdivision:
Total distance = 101.5 + 206.5 + 211 + 216 = 735 miles (approximate)
B. To find an approximation for S''(6), we can use the formula for the second derivative:
S''(6) = (S'(7) - 2S'(6) + S'(5)) / (1^2)
Using the given speed measurements, we can approximate S''(6):
S''(6) = (223 - 2 * 219 + 212) / 1 = 7 mph/min^2
This approximation represents the rate of change of speed with respect to time at the 6th minute. It indicates how quickly the car's speed is changing at that specific moment.
C. To find the average speed, we need to calculate the total distance traveled and divide it by the total time taken.
Total distance = 735 miles (from part A)
Total time = 12 minutes
Average speed = Total distance / Total time = 735 / 12 = 61.25 mph
The car's average speed over the first 12 minutes is approximately 61.25 mph.
To qualify for the race, the car needs to have an average speed of 210 mph. Since the car's average speed is only 61.25 mph, it is currently not fast enough to qualify. It needs to improve its speed significantly.
A. To approximate the total distance the car traveled over the first 12 minutes using the trapezoid rule with 4 equal subdivisions, we can use the following formula:
∆d = (t₂ - t₁) * (S(t₁) + S(t₂)) / 2
Where ∆d represents the change in distance, t₁ represents the initial time, t₂ represents the final time, and S(t₁) and S(t₂) represent the initial and final speeds, respectively.
Let's calculate the total distance traveled step by step:
∆d₁ = (1 - 0) * (S(0) + S(1)) / 2
∆d₂ = (2 - 1) * (S(1) + S(2)) / 2
∆d₃ = (3 - 2) * (S(2) + S(3)) / 2
∆d₄ = (4 - 3) * (S(3) + S(4)) / 2
Now, we can sum up the individual ∆d values to find the total distance:
Total distance = ∆d₁ + ∆d₂ + ∆d₃ + ∆d₄
Plugging in the given values from the table:
∆d₁ = (1 - 0) * (201 + 205) / 2 = 206.0 miles
∆d₂ = (2 - 1) * (205 + 208) / 2 = 206.5 miles
∆d₃ = (3 - 2) * (208 + 214) / 2 = 210.0 miles
∆d₄ = (4 - 3) * (214 + 218) / 2 = 216.0 miles
Total distance = 206.0 + 206.5 + 210.0 + 216.0 = 838.5 miles
Therefore, the car traveled approximately 838.5 miles over the first 12 minutes using the trapezoid rule with 4 equal subdivisions.
B. To find an approximation for S''(6), we can use the divided difference formula:
S''(6) = (S'(7) - S'(5)) / (t(7) - t(5))
Where S'(t) represents the first derivative of S with respect to time t.
Using the given table values:
S'(5) = (S(6) - S(4)) / (t(6) - t(4)) = (219 - 218) / (6 - 4) = 0.5 mph/min
S'(7) = (S(8) - S(6)) / (t(8) - t(6)) = (220 - 219) / (8 - 6) = 0.5 mph/min
S''(6) = (0.5 - 0.5) / (7 - 5) = 0.0 mph/min²
Therefore, the approximation for S''(6) is 0.0 mph/min². In the context of the problem, this quantity represents the acceleration of the car at 6 minutes. Since the second derivative is zero, it means that the car's acceleration is constant at that time, neither accelerating nor decelerating.
C. To find the car's average speed over the first 12 minutes, we can use the formula:
Average speed = Total distance / Total time
The total time is given by the difference between the last and first time values: 12 - 0 = 12 minutes.
Using the total distance calculated in part A as 838.5 miles:
Average speed = 838.5 miles / 12 minutes
Converting minutes to hours:
Average speed = 838.5 miles / (12/60) hours = 838.5 miles / 0.2 hours
Average speed = 4192.5 mph
The car's average speed over the first 12 minutes is approximately 4192.5 mph.
To qualify for the race, the car needs to have an average speed of 210 mph. Since the car's average speed over the first 12 minutes is well above 210 mph (4192.5 mph), it is currently fast enough to qualify for the race.
A. To approximate the total distance the car traveled using the trapezoid rule, we need to find the area under the curve of the speed-time function. The trapezoid rule divides the area into trapezoids and approximates each trapezoid's area as the average of the heights at the two endpoints, multiplied by the width (time interval) between those endpoints.
Let's first calculate the time interval, which is 12 minutes divided by the number of subdivisions (4 in this case). The time interval is 12/4 = 3 minutes.
Using the trapezoid rule, we calculate the area for each trapezoid and then add them up to get the total distance.
For the first trapezoid:
Average height = (S(0) + S(3))/2 = (201 + 214)/2 = 415/2 mph
Area of first trapezoid = Average height * time interval = (415/2) * 3 minutes
Similarly, you can calculate the areas for the other three trapezoids using the formula mentioned above.
For the second trapezoid:
Average height = (S(3) + S(6))/2
Area of second trapezoid = Average height * time interval
For the third trapezoid:
Average height = (S(6) + S(9))/2
Area of third trapezoid = Average height * time interval
For the fourth trapezoid:
Average height = (S(9) + S(12))/2
Area of fourth trapezoid = Average height * time interval
Finally, add up the areas of all four trapezoids to get the total distance the car traveled over the first 12 minutes.
B. To approximate S''(6), we need to find the second derivative of the speed function. However, we don't have the exact functional form of the speed function from the given measurements. Hence, we'll use a numerical approximation to the second derivative.
The second derivative approximated at a specific point can be found using central differences. The formula is:
S''(t) ≈ (S(t+h) - 2S(t) + S(t-h)) / h^2
Here, h is a small step size. Let's assume h = 1 minute.
Using the formula, we can approximate S''(6) as:
S''(6) ≈ (S(7) - 2S(6) + S(5)) / 1 minute^2
Substitute the values from the table into the formula to calculate the approximation for S''(6). Make sure to include the units (mph/minute^2) when reporting the value.
The quantity S''(6) represents the acceleration of the car at the 6th minute. It measures the rate at which the car's speed is changing. A positive value indicates an increase in speed, while a negative value indicates a decrease in speed.
C. To find the car's average speed over the first 12 minutes, we can use the average value of the speed function.
Average speed = (Sum of all speed measurements) / Number of measurements
Calculate the sum of all speed measurements given in the table, and divide it by the number of measurements (which is 12 in this case) to find the average speed over the first 12 minutes.
Compare the average speed to the required speed of 210 mph to determine if the car is fast enough to qualify for the race.