A cylinder has a surface area of 54π cm2 . If it were dilated using a scale factor of 12 , what is the surface area of the new cylinder?
1.)13.5π cm2
2.)6.75π cm2
3.)108π cm2
4.)27π cm2
To solve this problem, we need to understand how the surface area of a cylinder changes when it is dilated using a scale factor. The surface area of a cylinder is given by the formula:
SA = 2πr² + 2πrh
where r is the radius of the base and h is the height of the cylinder.
If we dilate the cylinder by a scale factor of 12, that means all of its dimensions increase by a factor of 12. The new radius, r', is equal to 12r, and the new height, h', is equal to 12h.
Plugging these new values into the surface area formula, we get:
SA' = 2π(12r)² + 2π(12r)(12h)
= 288πr² + 288πrh
We know that the original surface area of the cylinder is 54π cm². So, we can set up the equation:
54π = 288πr² + 288πrh
Simplifying the equation, we get:
r² + rh = 1/6
Now we can substitute the given choices for the surface area of the new cylinder and see which one satisfies the equation.
1.) 13.5π cm²:
r'² + r'h' = 1/6
(12r)² + (12r)(12h) = 1/6
144r² + 144rh = 1/6
This choice does not satisfy the equation.
2.) 6.75π cm²:
r'² + r'h' = 1/6
(12r)² + (12r)(12h) = 1/6
144r² + 144rh = 1/6
This choice does not satisfy the equation.
3.) 108π cm²:
r'² + r'h' = 1/6
(12r)² + (12r)(12h) = 1/6
144r² + 144rh = 1/6
This choice satisfies the equation.
4.) 27π cm²:
r'² + r'h' = 1/6
(12r)² + (12r)(12h) = 1/6
144r² + 144rh = 1/6
This choice does not satisfy the equation.
Therefore, the correct answer is:
3.) 108π cm²