   Chapter 16.7, Problem 40E

Chapter
Section
Textbook Problem

Find the mass of a thin funnel in the shape of a cone z   =   x 2 + y 2 , 1 ≤ z ≤ 4, if its density function is ρ(x, y, z) = 10 - z.

To determine

To find: The mass of a thin funnel in the shape of a cone.

Explanation

Given data:

ρ(x,y,z)=10z and z=x2+y2 , 1z4 .

Formula used:

m=Sρ(x,y,z)dS (1)

SFdS=DF(rϕ×rθ)dA (2)

rϕ=xϕi+yϕj+zϕk (3)

Substitute 10z for ρ(x,y,z) and x2+y2 for z in equation (1),

m=S(10x2+y2)dS

Modify equation (2).

S(10x2+y2)dS=D(10x2+y2)|rx×ry|dA (4)

Find rx .

Modify equation (3).

rx=xxi+yxj+zxk

Substitute x for x, y for y and x2+y2 for z,

rx=x(x)i+x(y)j+x(x2+y2)k=i+(0)j+(2x2x2+y2)k=i+(xx2+y2)k

Find ry .

Modify equation (3).

ry=xyi+yyj+zyk

Substitute x for x , y for y and x2+y2 for z ,

ry=y(x)i+y(y)j+y(x2+y2)k=(0)i+j+(2y2x2+y2)k=j+(yx2+y2)k

Find rx×ry .

rx×ry=(i+(xx2+y2)k)×(j+(yx2+y2)k)=|ijk10xx2+y201yx2+y2|={(0xx2+y2)i+(0yx2+y2)j+(10)k}=xx2+y2iyx2+y2j+k

Find |rx×ry| .

|rx×ry|=|xx2+y2iyx2+y2j+k|=(xx2+y2)2+(yx2+y2)2+(1)2=x2x2+y2+y2x2+y2+1=x2+y2x2+y2+1

Simplify the equation

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

Evaluate the expression sin Exercises 116. 230

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 1124, find the indicated limits, if they exist. 23. limx3x2+2x+42x23x+1

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Express each number in terms of j: 36

Elementary Technical Mathematics

Find the following products. 4i(1i)2

Trigonometry (MindTap Course List)

True or False: ln(a + b) = ln a + ln b.

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

Sometimes, Always, or Never: fxy = fyx.

Study Guide for Stewart's Multivariable Calculus, 8th 