Speedometer readings for a motorcycle at 12-seconds intervals are given in the table.

Question

| t(sec) | 0 | 12 | 24 | 36 | 48 | 60 |

| v(ft/sec) | 23 | 21 | 30 | 22 | 23 | 28 |

(a) Estimate the distance, in feet, traveled by the motorcycle during the time interval from 0 to 60 seconds using the velocities at the beginning of the time intervals.

(b) Give another estimate using the velocities at the end of the time periods.

Answer 1

a) Calculate the area of each rectangle using geometry.

b)

Same only shift the rectangles to be on the right side. The first dot (at t = 0) will not have a rectangle.

Answer 2

a. To estimate the distance traveled by the car using velocities at the beginning(left side) of intervals we'll simply find the length of the time interval and multiply it by the speed. For instance, the first time interval is 12 second long and the speed at the beginning of the time interval is 23 ft/sec. If you go 23 ft/s for 12 seconds, then you simply multiply the values to find you traveled an estimated 23x12 = 276 ft. We just continue doing this for each interval. Notice each time interval is 12 seconds long.

So we should have:

12x23 + 12x21 + 12x30 + 12x22 + 12x23 = 1428 total feet traveled

Often times when the time interval is the same, you'll see people factor out the common interval of 12 so you'll write:

12(23 + 21 + 30 + 22 + 23) = 1428 ft.

Notice when we estimate from the beginning(left) of each interval we don't use the final value in the table.

b. We'll do the same here, but use the end(right side) value of each interval. Thus we should end up with:

12(28 + 23 + 22 + 30 + 21) = 1488 ft.