Question

A cone has a diameter of 12 ft and a slant height of 20 ft. Explain whether tripling both dimensions would triple the surface area.

Answer 1

multiplying a dimension by 3 will correspond to a 3^2 multiplication of the area.

Answer 2

??????

Answer 3

hmmm how do you find the surface area of a cone anyway?

Answer 4

S.A= pi(radius squared) + pi(radius x length)

Answer 5

what is 3^2 ?

Answer 6

9

Answer 7

rightsurface area is multiplied by 9

Answer 8

so no then?

Answer 9

hmmm so... \(\textit{surface area of a cone}=\pi r{\color{brown}{ \sqrt{r^2+h^2}}}+\pi r^2\qquad
\begin{cases}
r=radius\\
{\color{brown}{ slant\ height}}
\end{cases}
\\ \quad \\
\textit{now multiplying both by 3, we get}
\\ \quad \\
\pi (3)r{\color{brown}{ (3)\sqrt{r^2+h^2}}}+\pi (3)r^2\implies 9\pi r\sqrt{r^2+h^2}+3\pi r^2
\\ \quad \\
3\left[ 3\pi r\sqrt{r^2+h^2}+\pi r^2 \right]\)

Answer 10

a triple the surface area of 3SA
would be \(\bf 3SA = 3\left[ \pi r\sqrt{r^2+h^2}+\pi r^2 \right]\ne 3\left[ 3\pi r\sqrt{r^2+h^2}+\pi r^2 \right]\)

Answer 11

lets look at a simple example. This is applicable to all 3 dimensional shapes

if we have a cube of side 2 cms S.A = 6*2^2 = 24 cm^2

if we triple the side lengths
SA = 6* 6^2 = 216


the ratio of the area s is not 3:1 its 216 ; 24 = 9:1

Answer 12

the ratio is the square of the ratio of side lengths

Answer 13

the answer is NO