   Chapter 3.9, Problem 20E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s.(a) At what rate is his distance from second base decreasing when he is halfway to first base?(b) At what rate is his distance from third base increasing at the same moment? (a)

To determine

To find: The rate at which his distance from the second base decreasing when he is halfway to the first base.

Explanation

Given:

Let given that a baseball diamond is a square with side 90 ft.

A batter hits the ball and the ball run towards first base with a speed of 24 ft/s, that is dxdt=24ft/s .

Formula used:

(1) Chain rule: dydx=dydududx

(2) Pythagorean Theorem.

Calculation:

Let x be the distance of the batter from the base point H, and y be the distance between the batter and the point from baseball at which it dropped.

Since both x and y changes with the time t

Therefore, x and y both are function of the time variable t

The position of the batter and the baseball as shown in the figure-1 given below

Apply Pythagoras Theorem in the right side Right angle triangle,

y2=(90x)2+902

Obtain the value of y when x=45,

y2=(9045)2+902=452+902=2,025+8,100=10,125

On further simplification the value of y is as follows.

y=10,125=45×45×5=455ft .

Obtain dydt when he is halfway to first base

(b)

To determine

To find: The rate at which his distance from the third base increasing when he is halfway to the first base.

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