Question

A tomato sauce company currently sells tomato sauce in a cylindrical can that has a radius of 5 cm and a height of 8cm. They have plans to use larger cans to sell tomato sauce in bulk, The new cylindrical cans are to be similar to the original cans and have a height of 40cm. If the original can holds 127picm^3 of tomato sauce, how much tomato sauce will the new can hold?

Answer 1

SOLUTION

Now, the radius r of the smaller cylinder = 5cm

The height of the smaller cylinder = 8cm

And the volume of the smaller cylinder

$V_1=127\pi cm^3$

Let the height of the bigger cylinder be H = 40cm,

The radius of the bigger cylinder be R = ?

and the volume of the bigger cylinder be

$V_2=?$

Let us find the radius of the bigger cylinder or new can.

Since both cylinders are similar, then ratios of their radii to their heights will be equal. That is

$\begin{gathered} \frac{h}{r}=\frac{H}{R} \\ \frac{8}{5}=\frac{40}{R} \\ 8R=40\times5 \\ 8R=200 \\ R=\frac{200}{8} \\ R=25cm \end{gathered}$

Also since they are similar,

Their ratios of their Volumes and radii are related as follows

$\begin{gathered} \frac{R^3}{r^3}=\frac{V_2}{V_1} \\ (\frac{R}{r})^3=\frac{V_2}{V_1} \end{gathered}$

So, we have

$\begin{gathered} (\frac{R}{r})^3=\frac{V_2}{V_1} \\ (\frac{25}{5})^3=\frac{V_2}{127\pi_{}} \\ 5^3=\frac{V_2}{127\pi_{}} \\ 125=\frac{V_2}{127\pi_{}} \\ V_2=125\times127\pi \\ V_2=15875\pi cm^3 \end{gathered}$

Hence the volume of the new can will be

$=15875\pi cm^3$