a Let's start by drawing the described cylinder on graph paper.
Volume
To calculate the volume of a cylinder, we have to multiply the area of the base with its height. Since the base of a cylinder is circular, we can find its area by multiplying the radius with π. From the exercise we know that the radius is 6 inches.
Now we can find the volume by multiplying the area of the base with the height.
Volume: 36π(9)=324π ≈ 1017.9 in^3
Surface Area
The surface area of the cylinder is the sum of its external faces. We already know the area of the base. However, the cylinder has two bases, which means we have to double this number.
bases: 2(36π)=72π in.^2
If we could roll out the side of the cylinder it would have the shape of a rectangle, where the cylinder's height is one dimension and the circumference of the base is the second dimension. The circumference is the product of the diameter and π.
The area of the cylinder's side is 12π(9)=108π. Finally, by adding all of the external faces we can calculate the total surface area.
72π+108π=565.5 cm^2
b If the linear scale factor is 3, the enlarged cylinder has sides that are 3 times longer than the small cylinder.
Linear scale factor=3
If we cube both sides of this equation, we can calculate the corresponding volume scale factor.
(Linear scale factor)^3=( 3)^3
⇓
Volume scale factor=27
The volume of the enlarged cylinder is 27 times greater than the original. Therefore, by multiplying the original cylinder's volume by 27 we can determine the volume of the enlarged cylinder.
Volume: 324π(27)= 8748π≈ 27 483 in.^3
