# Math Help - Max/min word problems!

1. ## Max/min word problems!

1) A sheet of paper for a poster contains 18 square cm. The margins at the top and bottom are 0.75 cm and at the sides 0.5 cm. What are the dimensions if the printed area is to be a maximum?

2) A cylindrical container with circular base is to hold 64 cubic centimeters. Find the dimensions so that the surface area will be a minimum when the cylinder is closed at the top.

2. ## Max and min

Hello qzno
Originally Posted by qzno
1) A sheet of paper for a poster contains 18 square cm. The margins at the top and bottom are 0.75 cm and at the sides 0.5 cm. What are the dimensions if the printed area is to be a maximum?
This is a very small poster - more like a postage stamp! Only 18 sq cm?

Suppose the width of the paper is $x$ cm. Then, using area = length x width, the height is $\frac{18}{x}$ cm.

The width of the printed area is 1.0 cm less than that of the paper, and its height is 1.5 cm less. So the width and height of the printed area are ...?... cm and ...?... (in terms of $x$).

Multiply these to get an expression for the printed area, $A$ sq cm, and then find the value of $x$ that makes $A$ a maximum.

(Answer: $x = 3\sqrt{3} = 5.12$ cm. So the dimensions are ...?)

2) A cylindrical container with circular base is to hold 64 cubic centimeters. Find the dimensions so that the surface area will be a minimum when the cylinder is closed at the top.
If the radius of the cylinder is $r$ cm, and the height $h$ cm, then the volume $V = 64 = \pi r^2 h$

So $h =$ ...?... (in terms of $r$)

The total surface area of the cylinder, $S = 2\pi r^2 + 2\pi rh =$ ...?... (in terms of $r$ only)

Now find the value of $r$ that makes $S$ a minimum.

(Answer: $r = \sqrt[3]{\frac{32}{\pi}} = 2.17$ cm. So the dimensions are ...?)

Can you complete these now?