A finch rides on the back of a Galapagos tortoise, which walks at the stately pace of 0.060 m/s. After 72 seconds the finch tires of the tortoise’s slow pace, and takes flight in the same direction for another 72 seconds at 12 m/s. What was the average speed of the finch for this 144-second interval?

The average speed of the finch is: 6.03 m/s

Explanation:

Recall that the average value of a function in a given interval is defined as the integral of the function (area under the curve the function describes) in that interval, divided by the length of the interval.

In this case, the function that describes the speed of the finch (in the vertical axis) as a function of time (in the horizontal axis), is a piecewise function that consists of two horizontal lines (one at y = 0.06 between 0 and 72 seconds in the horizontal axis, and the other one at y = 12 between 72 and 144 in the horizontal axis).

The area of such function is simply the addition of the areas of two rectangles, both of the same width (72 seconds) but of different heights (one 0.06 m/s and the other one 12 m/s)

Then these areas add up to give:

0.06 * 72 + 12 * 72 = 868.32 m

When we divide this by the length of the interval of integration (144 seconds) to calculate the average speed, we get:

average speed = 868.32/144 = 6.03 m/s