The dimensions of a cylinder are dilated by a scale factor of V. How will the change of dimensions affect the surface area of the cylinder?
A. The new surface area is reduced by 50% of the original.
B. The new surface area is reduced to 1/4 of the original.
C. The new surface area is reduced to 1/8 of the original.
D. The surface area is 2 times the original.
The Correct Answer and Explanation is :
The correct answer is D. The surface area is 2 times the original.
Explanation:
When the dimensions of a three-dimensional object, such as a cylinder, are scaled by a factor ( V ), it means that all linear dimensions (such as the height and the radius of the cylinder) are multiplied by this factor. For a cylinder, its surface area is calculated using both the lateral surface area and the areas of its two circular bases.
Original surface area formula:
The surface area ( A ) of a cylinder with radius ( r ) and height ( h ) is given by:
[
A = 2\pi r^2 + 2\pi rh
]
where:
- ( 2\pi r^2 ) is the area of the two circular bases, and
- ( 2\pi rh ) is the lateral surface area (the side area of the cylinder).
When the cylinder is dilated by a scale factor ( V ), both the radius ( r ) and the height ( h ) are multiplied by ( V ). Let’s calculate the new surface area after scaling.
New surface area after scaling:
After the scale factor ( V ), the new radius becomes ( r’ = Vr ) and the new height becomes ( h’ = Vh ).
Substitute these into the surface area formula:
[
A’ = 2\pi (Vr)^2 + 2\pi (Vr)(Vh)
]
[
A’ = 2\pi V^2 r^2 + 2\pi V^2 rh
]
[
A’ = V^2 \left(2\pi r^2 + 2\pi rh\right)
]
This shows that the new surface area ( A’ ) is multiplied by ( V^2 ). Therefore, if the scale factor ( V = 2 ), then the new surface area will be ( 2^2 = 4 ) times the original. If ( V = 0.5 ), the surface area would be reduced.
The problem doesn’t specify ( V ), but under typical scaling (e.g., doubling the dimensions), the surface area increases significantly, as described in option D.