Question

"If the volume of two cylinders are equal, then their surfaces areas must be equal." Do you agree or disagree with this statement? Why or Why not? What experiment would you devise to prove that you are correct?

Answer 1

Answer:

This statement is true, because the surface area and the volume depend on the same variables.

$V_{1} =V_{1} \implies (r_{1} =r_{2} ) \ and \ (h_{1} =h_{2} )$

Step-by-step explanation:

The volume of a cylinder is defined as

$V=\pi \times r^{2} \times h$

The surface area of a cylinder is defined as

$S=2 \pi r^{2} + 2\pi rh$

Notice that if two cylinder have the same volume that means the radius and height are also equal.

$V_{1} =V_{1} \implies (r_{1} =r_{2} ) \ and \ (h_{1} =h_{2} )$

Now, if you analyse the surface definition of a cylinder, you would find that it depends on the radius and the height, which means if these variables are equal, then the area surfaces are also equal, like in this case.

Therefore, this statement is true, because the surface area and the volume depend on the same variables.

Answer 2

Check the picture below, same volumes.

$\bf \textit{surface area of a cylinder}\\\\ S=2\pi r(h+r)\\\\ -------------------------------\\\\ \textit{for the short cylinder}\quad \begin{cases} r=3\\ h=10 \end{cases}\implies S=2\pi 3(10+3) \\\\\\ \boxed{S=78\pi }\\\\ -------------------------------\\\\ \textit{for the tall cylinder}\quad \begin{cases} r=2\\ h=22.5 \end{cases}\implies S=2\pi 2(22.5+2) \\\\\\ \boxed{S=98\pi }$