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The height in feet of a projectile as a function of time, in seconds, is given by π of π‘ equals negative 16π‘ squared plus 92π‘.
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Find the average rate of change of height with respect to time between one and 1.5 seconds.
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The key to answering this question is to realize that the height is being given as a function π of π‘, and we have a formula that allows us to calculate the average rate of change of a function.
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Take a function π of π₯, the average rate of change of that function from the closed interval π to π is π of π minus π of π all divided by π minus π.
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Now, of course, our function is π of π‘.
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So weβre going to rewrite this a little bit and say that the average rate of change of π is π of π minus π of π over π minus π.
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But actually, weβre told that weβre trying to find the rate of change of height between one and 1.5 seconds.
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And so weβre going to let π be equal to one and π be equal to 1.5.
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And so our average rate of change will be π of 1.5 minus π of one over 1.5 minus one.
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Now, in fact, we can simplify the denominator of our fraction to get 0.5.
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And it should be clear that the next step is to work out π of 1.5 and π of one.
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π at 1.5 is found by substituting π‘ equals 1.5 into our height function.
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Itβs negative 16 times 1.5 squared plus 92 times 1.5.
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And thatβs 102 or 102 feet.
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Then we repeat this process for π of one.
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This time, thatβs negative 16 times one squared plus 92, which is 76 or 76 feet.
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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.
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102 minus 76 is 26, so we have 26 divided by 0.5.
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But of course, dividing by 0.5 is actually the same as multiplying by two.
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So weβre actually working out 26 times two which is 52.
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Weβre finding the average rate of change of height with respect to time, where height is given in feet and time in seconds.
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And so the units here are feet per second.
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And so the average rate of change of height with respect to time is 52 feet per second.