First, find the surface area of the cylinder as it is.
Then double the radius and the height, and calculate the surface area again.
Then compare the two surface areas.
The surface area of a cylinder is the sum of the areas of the two bases and the lateral area.
[tex]\bf{surface ~area} = 2 \pi r^2 + 2 \pi rh[/tex]
Original dimensions: r = 3 cm; h = 5 cm
[tex]\bf{original~surface ~area} = 2 \pi (3~cm)^2 + 2 \pi (3~cm)(5~cm)[/tex]
[tex]\bf{original~surface ~area} = 48~cm^2[/tex]
Doubled dimensions: r = 6 cm; h = 10 cm
[tex]\bf{new~surface ~area} = 2 \pi (6~cm)^2 + 2 \pi (6~cm)(10~cm)[/tex]
Now we compare the new surface area to the original surface area.
[tex]\bf{new~surface ~area} = \dfrac{192~cm^2}{48~cm^2} = 4[/tex]
The linear dimensions were increased by a factor of 2.
The surface area was increased by a factor of 4.
4 is the same as [tex] 2^{2} [/tex]. When you increase the linear dimensions by a factor of k, you increase the area by a factor of [tex] k^{2} [/tex].
