For a cone, if the radius was quadrupled and the slant height was reduced to one sixth of its original size, what would be the formula to find the modified surface area?

surface area=PI*r*hs + PI*r^2

radius=4x hs=1/6 x

modified area=2PI*hs/3 + PI (16)r^2 where r, hs are the original slant heights, and radius.

To find the modified surface area of a cone, you need to consider the changes in radius and slant height.

The surface area of a cone can be calculated using the formula:
A = πr(r + √(r^2 + h^2))

Let's break down the changes and modify the formula accordingly:

1. Change in radius: The radius is quadrupled, which means it becomes four times its original size. So, the new radius would be 4r.

2. Change in slant height: The slant height is reduced to one-sixth of its original size. This means the new slant height is (1/6)h.

Now, substitute these changes into the original formula:

A = π(4r)[4r + √((4r)^2 + (1/6)h)^2]
= π(4r)[4r + √(16r^2 + (1/36)h^2)]

Simplifying further, we have:
A = 16πr^2 + πr√(16r^2 + (1/36)h^2)

Therefore, the formula to find the modified surface area of the cone with a quadrupled radius and reduced slant height would be:
Modified Surface Area = 16πr^2 + πr√(16r^2 + (1/36)h^2)