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

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} (h)/(r)=(H)/(R) \\ (8)/(5)=(40)/(R) \\ 8R=40*5 \\ 8R=200 \\ R=(200)/(8) \\ R=25cm \end{gathered}`

Also since they are similar,

Their ratios of their Volumes and radii are related as follows

`\begin{gathered} (R^3)/(r^3)=(V_2)/(V_1) \\ ((R)/(r))^3=(V_2)/(V_1) \end{gathered}`

So, we have

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

Hence the volume of the new can will be

`=15875\pi cm^3`