The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table below. Find lower and upper estimates of the distance she covered during the first three seconds of the race. Use left- and right- rectangles. Do not use midpoint rectangles. In order to get the best estimates, use 0.5 rectangle widths.
Enter your answer as a Riemann sum to show your work.
The lower estimate is:
The upper estimate is:
t(sec) 0 0.5 1 1.5 2 2.5 3
v(ft/sec)0 6.8 12.4 16.4 19.5 23 23.8
To find the lower and upper estimates of the distance covered during the first three seconds of the race using left- and right- rectangles, we can break the time interval into smaller subintervals and compute the area of rectangles for each subinterval.
The given table provides the speed of the runner at half-second intervals. To calculate the lower and upper estimates, we will use rectangle widths of 0.5 seconds.
Lower Estimate:
To calculate the lower estimate, we will use left rectangles. This means we will use the speed value in each subinterval to approximate the distance covered.
Subinterval 1: 0 to 0.5 seconds
The speed is given as 0 ft/sec.
Area of the rectangle = speed × width = 0 ft/sec × 0.5 sec = 0 ft
Subinterval 2: 0.5 to 1 second
The speed is given as 6.8 ft/sec.
Area of the rectangle = speed × width = 6.8 ft/sec × 0.5 sec = 3.4 ft
Subinterval 3: 1 to 1.5 seconds
The speed is given as 12.4 ft/sec.
Area of the rectangle = speed × width = 12.4 ft/sec × 0.5 sec = 6.2 ft
Subinterval 4: 1.5 to 2 seconds
The speed is given as 16.4 ft/sec.
Area of the rectangle = speed × width = 16.4 ft/sec × 0.5 sec = 8.2 ft
Subinterval 5: 2 to 2.5 seconds
The speed is given as 19.5 ft/sec.
Area of the rectangle = speed × width = 19.5 ft/sec × 0.5 sec = 9.75 ft
Subinterval 6: 2.5 to 3 seconds
The speed is given as 23.8 ft/sec.
Area of the rectangle = speed × width = 23.8 ft/sec × 0.5 sec = 11.9 ft
Now, let's sum up the areas of all the rectangles to get the lower estimate:
Lower Estimate = 0 + 3.4 + 6.2 + 8.2 + 9.75 + 11.9 = 39.45 ft
So, the lower estimate of the distance covered during the first three seconds of the race is 39.45 feet.
Upper Estimate:
To calculate the upper estimate, we will use right rectangles. This means we will use the speed value at the end of each subinterval to approximate the distance covered.
Subinterval 1: 0 to 0.5 seconds
The speed at the end of the subinterval is given as 6.8 ft/sec.
Area of the rectangle = speed × width = 6.8 ft/sec × 0.5 sec = 3.4 ft
Subinterval 2: 0.5 to 1 second
The speed at the end of the subinterval is given as 12.4 ft/sec.
Area of the rectangle = speed × width = 12.4 ft/sec × 0.5 sec = 6.2 ft
Subinterval 3: 1 to 1.5 seconds
The speed at the end of the subinterval is given as 16.4 ft/sec.
Area of the rectangle = speed × width = 16.4 ft/sec × 0.5 sec = 8.2 ft
Subinterval 4: 1.5 to 2 seconds
The speed at the end of the subinterval is given as 19.5 ft/sec.
Area of the rectangle = speed × width = 19.5 ft/sec × 0.5 sec = 9.75 ft
Subinterval 5: 2 to 2.5 seconds
The speed at the end of the subinterval is given as 23 ft/sec.
Area of the rectangle = speed × width = 23 ft/sec × 0.5 sec = 11.5 ft
Subinterval 6: 2.5 to 3 seconds
The speed at the end of the subinterval is given as 23.8 ft/sec.
Area of the rectangle = speed × width = 23.8 ft/sec × 0.5 sec = 11.9 ft
Now, let's sum up the areas of all the rectangles to get the upper estimate:
Upper Estimate = 3.4 + 6.2 + 8.2 + 9.75 + 11.5 + 11.9 = 51.95 ft
So, the upper estimate of the distance covered during the first three seconds of the race is 51.95 feet.