Question

A cylindrical container of height equal to twice the diameter of its base can

hold 12 liters (1L= 1,000 cm3) of water. Another cylindrical container with

the same capacity has its height equal to three times the diameter of its

base.


1.What is the diameter of the second container?

Answer 1

Let $d_1=$ the diameter of the base of the first container, $h_1=$ the heigth of the first container and $V_1=$ the volume of the first container.

Let $d_2=$ the diameter of the base of the second container, $h_2=$ the heigth of the second container and $V_2=$ the volume of the second container.

Then

$h_1=2d_1, V_1=\pi (\dfrac{d_1}{2})^2h_1=\dfrac{1}{2}\pi d_1^3=\dfrac{1}{16}\pi h_1^3$

$h_2=3d_2, V_1=\pi (\dfrac{d_2}{2})^2h_2=\dfrac{3}{4}\pi d_2^3=\dfrac{1}{36}\pi h_2^3$

Given

$$V_1=V_2=12L=12000cm^3$$

$$\dfrac{3}{4}\pi d_2^3=12000cm^3$$

$$d_2=\sqrt[3]{\dfrac{4(12000)}{3\pi}}cm$$

$$d_2=20\sqrt[3]{\dfrac{2}{\pi}}cm\approx 17.205\ cm$$

The diameter of the second container is $20\sqrt[3]{\dfrac{2}{\pi}}cm\approx 17.205\ cm.$