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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

If a diver of mass m stands at the end of a diving board with length L and linear density ρ , then the board takes on the shape of a curve y = f ( x ) , where

E I y = m g ( L x ) + 1 2 ρg ( L x ) 2

E and I are positive constants that depend on the material of the board and g ( <   0 ) is the acceleration due to gravity.

(a) Find an expression for the shape of the curve.

(b) Use f(L) to estimate the distance below the horizontal at the end of the board.

Chapter 3.9, Problem 64E, If a diver of mass m stands at the end of a diving board with length L and linear density , then the

To determine

a)

To find:

An expression for the shape of the curve.

Explanation

1) Concept:

Theorem: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is Fx+c, where c is an arbitrary constant.

2) Given:

EIy''=mgL-x+12L-x2

E and I are the positive constants that depend on the material of the board, and

g<0  is the acceleration due to gravity.

3) Calculation:

Here, a diver of mass m stands at the end of a diving board with length L and linear density p, and then the board takes on the shape of a curve y=fx.

EIy''=mgL-x+12L-x2

Taking the general antiderivative by using the rules of antiderivative is

EIy'=-12mgL-x2-16pgL-x3+C , where C as a constant.

Again taking the antiderivative,

EIy=16mgL-x3+124pgL-x4+Cx+D ,  where D as a constant.

At the left end of board we have, y=y'=0 when x=0

So, substitute x=0 in y and y' to get C and D,

0=-12mgL-02-16pgL-03+C

C=12mgL2+16pgL30=-16mgL-03+124pgL-04+C

To determine

b)

To estimate:

The distance below the horizontal at the end of the board.

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