Question

A right cone has radius 9cm and slant height 12cm. The radius and slant height are both multiplied by 2/3. Which of the following correctly describes the effect on the surface area?
The surface area is multiplied by 4/9.

The surface area is multiplied by 2/9.

The surface area is multiplied by 4/3.

The surface area is multiplied by 2/3.

Answer 1

Answer:

The surface area is multiplied by 4/9.

Step-by-step explanation:

Surface Area of a Cone

Being r the radius and l the slant height of a cone, then its surface area is

$SA=\pi r^2+\pi r l$

Now let's think both the radius and the slant height are multiplied by the same factor k, i.e.

r'=kr

l'=kl

The new surface area is

$SA'=\pi r'^2+\pi r' l'=\pi (kr)^2+\pi \cdot kr\cdot kl=k^2\pi r^2+k^2\pi rl$

$SA'=k^2(\pi r^2+\pi rl)=k^2\cdot SA$

The new surface area is the  $k ^2$ times original SA. Since k=2/3, then

$SA'=4/9\ SA$

The surface area is multiplied by 4/9.