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?
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To find the modified surface area of the cone after quadrupling the radius and reducing the slant height to one-sixth of its original size, we need to understand the surface area formula for a cone.
The surface area formula for a cone is given by:
A = πr(r + l)
Where:
A = Surface area of the cone
r = Radius of the base
l = Slant height
To find the modified surface area, we need to substitute the modified values into this formula.
Let's denote the original radius as r₀ and the original slant height as l₀.
After quadrupling the radius, the new radius (r') becomes 4r₀, and the reduced slant height is one-sixth of the original slant height, so the new slant height (l') becomes l₀/6.
Substituting these modified values into the surface area formula, we have:
A' = π(4r₀)(4r₀ + l₀/6)
Simplifying the equation further:
A' = π(16r₀² + l₀/6)
Therefore, the formula to find the modified surface area (A') of the cone is A' = π(16r₀² + l₀/6).