   Chapter 14.6, Problem 30E

Chapter
Section
Textbook Problem

Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + 0.02x2 − 0.00 ly3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point (80. 60) and moves toward the buoy, which is located at (0, 0). Is the water under the boat getting deeper or shallower when he departs? Explain.

To determine

To explain: Whether the water under the boat getting deeper or shallower when he departs.

Explanation

Result used:

“The directional derivative of the function f(x,y,z) at f(x0,y0,z0) in the direction of unit vector u=a,b,c is Duf(x,y,z)=f(x,y,z)u , where f(x,y,z)=fx,fy,fz=fxi+fyj+fzk .”

Given:

The depth of a lake at the point (x,y) is z=200+0.02x20.001y3 where x, y, z are measured in meters.

The buoy starts at the point (80,60) and moves towards a point (0,0) .

Calculation:

The directional derivative is defined as, Duz(x,y)=z(x,y)u . (1)

Find the value of z(x,y) .

z(x,y)=zx,zy=x(200+0.02x20.001y3),y(200+0.02x20.001y3)=(0.02(2x)),(0.001(3y2))=(0.04x),(0.003y2)

Thus, the value of z(x,y)=(0.04x),(0.003y2) .

Substitue (x,y)=(80,60) in z(x,y) and compute z(80,60) as follows

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find all possible real solutions of each equation in Exercises 3144. y32y22y3=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the limit. limr9r(r9)4

Single Variable Calculus: Early Transcendentals, Volume I

Solve each equation: c2+2=3c

Elementary Technical Mathematics

For f (x) = sinh x, f (x) = a) cosh2 x b) sinh2 x c) tanh x d) sinh x

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 