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Chapter 4: Q. 56 (page 236)

A steel drum in the shape of a right circular cylinder is required to have a volume of 100 cubic feet.

(a) Express the amount A of material required to make the drum as a function of the radius r of the cylinder.

(b) How much material is required if the drum鈥檚 radius is 3 feet?

(c) How much material is required if the drum鈥檚 radius is 4 feet?

(d) How much material is required if the drum鈥檚 radius is 5 feet?

(e ) Graph A=A(r). For what value of r is A smallest?

Short Answer

Expert verified

Part (a) The amount of material required to make the drum is A(r)=200r+2蟺谤2.

Part (b) A(3)=2003+18蟺

Part (c) A(4)=50+32蟺

Part (d) A(5)=40+50蟺

Part (e) A is minimum atr=2.51.

Step by step solution

01

Part (a) Step 1. Use the formula for the volume of the cylinder.

Volume of the cylinder =蟺谤2h

蟺谤2h=100h=100蟺谤2

Now the total surface area of the cylinder is

A=2蟺谤2+2蟺谤hA=2蟺谤2+2蟺谤+(100蟺谤2)A=2蟺谤2+200r

or A(r)=200r+2蟺谤2

The material A used to make drum as a function of radius r of the cylinder is200r+2蟺谤2.

02

Part (b) Step 1. Substitute r=3 in A(r)=200r+2πr2.

This gives

A(3)=2003+2蟺(32)A(3)=2003+18蟺

03

Part (c) Step 1. Substitute r=4 in A(r)=200r+2πr2.

This gives

A(4)=2004+2蟺(42)A(4)=50+32蟺

04

Part (d) Step 1. Substitute r=5 in A(r)=200r+2πr2.

This gives

A(5)=2005+2蟺(52)A(5)=40+50蟺

05

Part (e) Step 1. The graph of A(r)=200r+2πr2 is

From the graph,

A is minimum atx=2.51.

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