How Do You Maximize the Surface Area of a Cylinder Using Rectangular Sheets?

[scientist]

May I have a tutor check over my question and solution?

Problem

Two rectangular sheets 20cm x 24cm and 15cm x 30cm are to be rolled to form cylinders. What is the height and diameter of the cylinder with maximum surface area that can be formed using either of these sheets?

My Solution

The surface area of a cylinder, including the ends, is 2* (pi)*r*h + 2 *(pi) *r^2. I want r to be the largest value. So for the first sheet, r = 24cm and h = 20cm and d = 12cm.

For the second sheet, r = 30cm, h = 15cm, and d = 15cm.

I welcome any comments or corrections.

 

[daveb]

Your reasoning is sound in that the cylindrical part of the cylinder will have the same area regardles of the orientation of the sheet, so you want to maximize the area of the two ends. However, your radius isn't 24cm (and 30cm) since that will end up being the circumference once you roll it up. You want to find the diamter based on the circumferences of 24cm and 30 cm.

 

[scientist]

Your reasoning is sound in that the cylindrical part of the cylinder will have the same area regardles of the orientation of the sheet, so you want to maximize the area of the two ends. However, your radius isn't 24cm (and 30cm) since that will end up being the circumference once you roll it up. You want to find the diamter based on the circumferences of 24cm and 30 cm.

------------------------------------------------------

Would then the diameter be this:

diameter = circumference / pi

= 24cm
------
3.14

= 7.64cm
For the first sheet the diameter is 7.64cm.


diameter = circumference / pi

= 30cm
--------
3.14

= 9.55cm
For the second sheet the diameter is 9.55cm.

Is this more correct?

 

[daveb]

Absolutely!