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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the Wood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point P(x, y) on the surface of seawater is approximated by

C(x, y) = e−(x2 + 2y2)/10x

where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin.

(a) Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source.

(b) Suppose a shark is at the point (x0, y0) when it first detects the presence of blood in the water. Find an equation of the shark's path by setting up and solving a differential equation.

(a)

To determine

To identify: The level curves of the concentration function C(x,y)=e(x2+2y2)104 use it to sketch a path that a shark will follow to the source.

Explanation

Given:

The concentration of blood at a point P(x,y) on the surface of seawater is approximately

C(x,y)=e(x2+2y2)104 .

Where xandy are measured in meters in rectangular coordinate system with source at origin.

Calculation:

The level curves of a function C(x,y) are the curves with equations, C(x,y)=k where k is a constant. Thus, e(x2+2y2)104=k .

It can be simplified as follows,

e(x2+2y2)104=k

Take ln on both side,

ln[(x2+2y2)104]=lnkln((x2+2y2))ln(104)=lnkln((x2+2y2))=lnk+ln(104)ln((x2+2y2))=ln(10

(b)

To determine

To find: The equation of shark’s path by using differential equation.

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Chapter 14 Solutions

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