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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Another reasonable model for the water speed of the river in Exercise 33 is a sine function: f(x) = 3 sin(πx/40). If a boater would like to cross the river from A to B with constant heading and a constant speed of 5 m/s, determine the angle at which the boat should head.

To determine

To find: The angle at which the boat should head.

Explanation

Given data:

f(x)=3sin(πx40) , x=40 and v0=5ms .

Formula used:

Write the expression to find the velocity.

v(t)=r(t)

r(t)=v(t)dt (1)

Let α be the angle north of east that the boat heads.

The velocity of the boat is v .

v(t)=v0cosαi+v0sinαj

Substitute 5 for v0 ,

v=5(cosα)i+5(sinα)j

At t seconds, the boat is 5(cosα)t meters from the west bank, at which point the velocity of the water is.

f(x)=3sin(πx40)j

Substitute 5(cosα)t for x ,

f(x)=3sin[π(5(cosα)t)40]j=3sin(π8tcosα)j

The resultant velocity of the boat is v(t) .

v(t)=v+f(x)

Substitute 5(cosα)i+5(sinα)j for v and 3sin(π8tcosα)j for f(x) ,

v(t)=5(cosα)i+5(sinα)j+3sin(π8tcosα)j=5(cosα)i+[5(sinα)+3sin(π8tcosα)]j

Substitute 5(cosα)i+[5(sinα)+3sin(π8tcosα)]j for v(t) in equation (1),

r(t)={5(cosα)i+[5(sinα)+3sin(π8tcosα)]j}dt=5(cosα)idt+5(sinα)jdt+3sin(π8tcosα)jdt=(5tcosα)i+(5tsinα)j24πcosαcos(π8tcosα)j+C

r(t)=(5tcosα)i+[(5tsinα)24πcosαcos(π8tcosα)]j+C (2)

If we place the boat initially at origin that is r(0)=0 and t=0

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