Rocket A has a positive velocity v(t) after being launched upward from an initial height of 0 feet at t=0 seconds. The velocity of the rocket is recorded for certain values of t over the interval 0 <= t <= 80 seconds, as shown below.
t(seconds),v(t)(ft/sec) = (0,5),(10,14),(20,22),(30,29),(40,35),(50,40),(60,44),(70,47),(80,49)
Use right riemann sums and three equal intervals to find the average value of the function on the interval [0,70].
I made three intervals from 10 to 30, 30 to 50, and 50 to 70. Taking the values on the right (30,50,70) I made a riemann sum as 20(v(30) + v(50) + v(70)) and got 2320.
2320/60=39.3
good numbers, but what about [0,10]?
Oops, the question asks to find the average on [10,70].
To find the average value of the function using right Riemann sums and three equal intervals, you need to follow these steps:
1. Calculate the width of each interval by subtracting the starting point from the ending point and dividing it by the number of intervals. In this case, the interval is [0,70], and you've divided it into three equal intervals, so each interval has a width of (70-0)/3 = 70/3.
2. Determine the right endpoints of each interval. You correctly identified the right endpoints as 30, 50, and 70.
3. Evaluate the function at the right endpoints of each interval. Using the given velocity values, the function evaluated at these points would be v(30) = 29 ft/sec, v(50) = 40 ft/sec, and v(70) = 47 ft/sec.
4. Multiply each function value by the width of the interval. In this case, each interval has a width of 70/3, so you would multiply each function value by 70/3.
5. Sum up the products obtained from step 4.
So, the average value of the function on the interval [0,70] using right Riemann sums and three equal intervals would be:
(70/3) * [v(30) + v(50) + v(70)]
= (70/3) * (29 + 40 + 47)
= (70/3) * 116
= 2320/3
≈ 773.33 ft/sec.