Question

Suppose you are required to design a rectangu-lar medicine container such that the ratio of thelength to the width to the height is 5:3:2,and the volume of the container equals120 cm?, how would you determine the dimen-sions (length, width, height) of this container?

Answer 1

For this case we know that we need to design a rectangular container and we have the following ratio:

lenght:width:height (5:3:2)

We also know that the total volume is given by 120 cm^3.

And we want to determine the dimensions of the container.

Step 1: Formula for the volume

Since we have a rectangular container the volume would be given by:

$V=l\cdot w\cdot h$

Where l= legth, w= width and h=height

Step 2 : Set up the formulas to use

If we select a dimension fix for example the lenght we can do the following:

$V=l(\frac{3}{5}l)(\frac{2}{5}l)$

The reason of this is because we have that:

$\frac{l}{w}=\frac{5}{3}\rightarrow w=\frac{3}{5}l$$\frac{l}{h}=\frac{5}{2}\rightarrow h=\frac{2}{5}l$

Using the formula of the volume with l we can do this:

$120\operatorname{cm}^3=\frac{6}{25}l^3$

Step 3: Solving for the answers

And solving for l we got:

$l=\sqrt[3]{\frac{25}{6}(120)}=7.937\operatorname{cm}$

And we can find the other dimensions like this:

$w=\frac{3}{5}l=\frac{3}{5}(7.937)=4.762\operatorname{cm}$$h=\frac{2}{5}l=\frac{2}{5}(7.937cm)=3.1748\operatorname{cm}$