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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 ρ g ( 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.

(a)

To determine

To find: The expression for the shape of curve.

Explanation

Given data:

Consider the expression for shape of board.

y=f(x) (1)

And

EIy=mg(Lx)+12ρg(Lx)2 (2)

Here,

E and I are constants,

m is mass,

L is length,

x is parameter, and

g is acceleration due to gravity.

Formula Used:

Antiderivative of (ax) is –1, xn is xn+1n+1 and 1 is x.

Calculation:

Antiderivate the equation (2) with respect to x.

EIy=mg((Lx)1+11+1)(1)+12ρg((Lx)2+12+1)(1)+C=12mg(Lx)212ρg((Lx)33)+C

EIy=12mg(Lx)216ρg(Lx)3+C (3)

Antiderivate the expression with respect to x.

EIy=12mg((Lx)2+12+1)(1)16ρg((Lx)3+13+1)+Ct+D=12mg((Lx)33)+16ρg((Lx)44)+Ct+D

EIy=16mg(Lx)3+124ρg(Lx)4+Cx+D (4)

In order to find the values of constants C and D, make the x, y and y equal to 0

(b)

To determine

To estimate: The distance between below horizontal at the end of board.

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