   Chapter 8.1, Problem 41E

Chapter
Section
Textbook Problem

A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180 − x 2 45 until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

To determine

The distance traveled by the prey from the time it is dropped until the time it hits the ground.

Explanation

Given information:

The parabolic trajectory of the falling prey is y=180x245 (1)

The prey hits the ground at y=0.

Substitute 0 for y in Equation (1).

0=180x245x245=180x2=8100x=90m

Therefore, the lower limit is 0 and the upper limit is 90 m.

The expression to find the distance traveled (L) is shown below:

L=ab1+(dydx)2dx (2)

Here, the derivative of the function y is dydx, the lower limit is a, and the upper limit is b.

Differentiate Equation (1) with respect to x.

dydx=2x45=2x45

Substitute 2x45 for dydx, 0 for a, and 90 m for b in Equation (2).

L=0901+(2x45)2dx=0901+4x2452dx (3)

Let u=245x (4)

Differentiate Equation (4).

du=245dx452du=dx

Substitute u for 245x and 452du for dx in Equation (3)

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