Gandalf took a piece of paper 5'' by 12'' and made a cylindrical can with it. He then put a top and bottom on the can and computed the can's surface area. Frodo did the same with another piece of 5'' by 12'' paper getting a different solution. Compute the surface area of both cans.
clearly, the lateral area is 5*12 = 60 in^2 in both cases
so how to get a different area? The top and bottom must be circles of different sizes
There are only two ways to curl up the sheet into a cylinder. What are they?
And recall that the area of a circle with circumference C is
A = C^2/(4π)
A = 2(pi*r^2) + pi*2r*h
A1 = 2(3.14*2.5^2) + 3.14*5*12 = 228 in^2.
A2 = 2(3.14*6^2) + 3.14*12*5 = 415 in^2.
To compute the surface area of the cans, we need to find the lateral surface area of each cylindrical can, as well as the areas of the top and bottom circles.
The lateral surface area of a cylinder can be calculated by multiplying the height of the cylinder by the circumference of its circular base. In this case, since the paper is used to create the height of the cylinder, the height is 12 inches.
The circumference of a circle can be found using the equation: circumference = 2πr, where r is the radius of the circle. In this case, the radius is half the width of the paper, which is 5 inches. So, the circumference of the circular base is 2π(5) = 10π inches.
Thus, the lateral surface area of each cylindrical can is given by multiplying the height (12 inches) by the circumference of the base (10π inches):
Lateral surface area = 12 * 10π = 120π square inches.
The top and bottom of each can are circles, and their areas can be found using the equation: area = πr^2, where r is the radius.
For the first can, the radius is 5 inches, so the area of each circle (top and bottom) is:
Area = π(5)^2 = 25π square inches.
For the second can, we were not given the dimensions of the paper, so let's assume it's similar (5'' by 12'') and follow the same calculations. Thus, the area of the top and bottom circles will also be 25π square inches.
Finally, we can compute the total surface area of each can by summing the areas of the lateral surface and the top/bottom circles:
First can:
Total surface area = Lateral surface area + 2*Area of top/bottom
= 120π + 2(25π)
= 120π + 50π
= 170π square inches.
Second can (assuming the same dimensions as the first can):
Total surface area = Lateral surface area + 2*Area of top/bottom
= 120π + 2(25π)
= 120π + 50π
= 170π square inches.
Therefore, both cans have a total surface area of 170π square inches.