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Chapter 11: Q. 86 (page 774)

Constructing a Cylindrical Tube: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Short Answer

Expert verified

If the tube is to have a volume of 224 cubic centimeters, then the size cardboard should be started with is13.02cm脳16.59cm.

Step by step solution

01

Step 1. Finding system of equations.

Let x and y denote the dimensions of the piece of cardboard and r denote the radius of rectangular tube.

As the area of the rectangle is 216cm2, then

xy=216

Next the volume of the tube is 224cm3which gives

蟺谤2y=224.

The rectangular tube is made by joining together two sides of rectangle cardboard which gives

x=2蟺谤x2蟺=r

So, we have

蟺(x2蟺)2y=224x2y4蟺=224

Thus the system of equations is

xy=216x2y4蟺=224

02

Step 2. Solve the system of equations.

From first equation,

xyx=216xy=216x

From second equation,

x2y4蟺=224x2(216x)4蟺=224216x4蟺=224x=896蟺216x=13.02cm

and

y=216xy=21613.02y=16.59cm

The required size of the cardboard is13.02cm脳16.59cm.

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