The height in feet of a projectile as a function of time, in seconds, is given by 𝑠 of 𝑡 equals negative 16𝑡 squared plus 92𝑡. Find the average rate of change of height with respect to time between one and 1.5 seconds.
The key to answering this question is to realize that the height is being given as a function 𝑠 of 𝑡, and we have a formula that allows us to calculate the average rate of change of a function. Take a function 𝑓 of 𝑥, the average rate of change of that function from the closed interval 𝑎 to 𝑏 is 𝑓 of 𝑏 minus 𝑓 of 𝑎 all divided by 𝑏 minus 𝑎. Now, of course, our function is 𝑠 of 𝑡. So we’re going to rewrite this a little bit and say that the average rate of change of 𝑠 is 𝑠 of 𝑏 minus 𝑠 of 𝑎 over 𝑏 minus 𝑎.
But actually, we’re told that we’re trying to find the rate of change of height between one and 1.5 seconds. And so we’re going to let 𝑎 be equal to one and 𝑏 be equal to 1.5. And so our average rate of change will be 𝑠 of 1.5 minus 𝑠 of one over 1.5 minus one. Now, in fact, we can simplify the denominator of our fraction to get 0.5. And it should be clear that the next step is to work out 𝑠 of 1.5 and 𝑠 of one. 𝑠 at 1.5 is found by substituting 𝑡 equals 1.5 into our height function. It’s negative 16 times 1.5 squared plus 92 times 1.5. And that’s 102 or 102 feet. Then we repeat this process for 𝑠 of one. This time, that’s negative 16 times one squared plus 92, which is 76 or 76 feet.
And so we then find that the average rate of change of height with respect to time is 102 minus 76 all over 0.5. 102 minus 76 is 26, so we have 26 divided by 0.5. But of course, dividing by 0.5 is actually the same as multiplying by two. So we’re actually working out 26 times two which is 52. We’re finding the average rate of change of height with respect to time, where height is given in feet and time in seconds. And so the units here are feet per second. And so the average rate of change of height with respect to time is 52 feet per second.