The height and radius of a cone are multiplied by 5. What effect does this have on the surface area of the cone?

The effect is that the new surface area is 5 times bigger when the radius and height are each multiplied by 5.

Step-by-step explanation:

The formula for surface area of a cone is given by the expression;

S = πrL

Where;

r is radius

L is slant height.

Now, in a cone, we know that the radius, vertical height and slant height form a triangle.

Thus, if vertical height is represented by h, using Pythagoras theorem, we will get;

h² = r² + L²

So, L² = h² - r²

L = √ (h² - r²)

So surface area of a cone would be;

S = πrL = πr (√ (h² - r²))

Now, we are told that the height and radius of a cone are each multiplied by 5.

Thus; r = 5r and h = 5h

So, our surface area is now;

S_new = πr (√ ((5h) ² - (5r) ²))

S_new = πr (√ ((25h²) - (25r²)))

Collecting like terms, we have;

S_new = πr (√ (25 (h² - r²)))

S_new = 5πr (√ (h² - r²))

Comparing our new surface area to the initial one gotten, it is clear that the new surface area is 5 times bigger when the radius and height are each multiplied by 5.