Change in elevation The elevation h (in feet above the ground)...

Question

Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).

Accepted Answer

Welcome back, everyone. A diver jumps from a cliff into the sea, and the height H and feet of the diver above sea level T seconds after jumping is given by the equation H of T equals 50 minus 16 T plus 4 T squared. Estimate the change in the diver's height over the interval, where T being between 1 and 1.6 inclusive. We're given 4 answer choices A 4.8 ft, B negative 4.8 ft, C 2.4 ft, and D negative 2.4 ft. So what we're going to do is simply use an approximation. The change in height H can be approximated as the derivative of the function H. At the initial point, our independent variable is T, so we're saying H at point C1, multiplied by the change in time delta C. So what we have to do is first we'll identify the derivative. We are finding the derivative of 50 minus 16 T + or T2. Here we have to apply the power rule, so we're going to get -16 + 8T. Our next goal is to valid H at T1. Our T1 based on the given interval is. One So essentially we're evaluating each at 1, which gives us -16 + 8 multiplied by 1, and this is simply negative 8. And now we want to identify delta H. Delta T, I'm where I write the change in time. So this is our T2 minus T1. Notice that our T2 is 1.6 and our T1 is 1. So we get 1.6 minus 1, which is 0.6. And now delta H can be approximated as H at T1, which is -8. Multiplied by 0.6. Now what are the units? Now the H prime is the derivative of H, right? So the rate of change of position, so that'd be feet per second. Which represents velocity and we're multiplying that by the change in time in seconds. So we're multiplying by 0.6 seconds. When we multiply these numbers, we're going to get negative 4.8 ft. Our units cancel out, right? We can cancel out the seconds, and essentially we have arrived at the final approximation. So that'd be -4.8 ft, which corresponds to the answer choice B. Thank you for watching.

Key Concepts

Derivative

Change in Function Value

Interval Notation

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