Volume of Smaller Barrel

Question

Two barrels filled with juice are similar in shape. The smaller barrel has a height of 3 feet. The larger barrel has a height of 5 feet. The total volume of the barrels is 456 cubic feet. What is the volume of the smaller barrel?

Answer 1

I had to really slow down and thoroughly think everything through in the last paragraph. This is actually a proportion problem. The fraction 27/125 is a scale factor for the volumes. (Adding the small and large parts of the ratio is a mistake.)

I like to identify the ratio parts as a fraction: S/L (small / large). The S/L just makes it easier for me to keep track of where which numbers (and the X) should go as I build that final equation.

Then the ratio is S/L = 27/125. We know the large volume and we need the small one. Just set a new fraction S/L = 27/125 = X/456.

I get X = 27*456/125 = 98.496.

Answer 2

The trick to solving this problem is knowing how the relationships between similar shapes work when you add dimensions.

Since the height of these cylinders is a one-dimensional measurement, the ratio between them will simply be a:b- or 3:5 in this specific case.

When you start talking about two-dimensional scale factors, the numbers become squared; for example, if you wanted to find the surface area of these cylinders, the ratio would be a^2 : b^2. In this case, it would be 9:25.

I'm sure you can already start to see the pattern for scale factors in higher dimensions. Finding the 3-dimensional measure of volume's scale factor, you would use a^3:b^3, which in this case would give you 27:125.

That means that, in our 456 cubic feet of volume, there exists a ratio of 27:125 divided into two cylinders. We can then turn the scale factor into parts of a whole; the small cylinder has 27/152 of the volume, and the larger cylinder has 125/152 of the volume (152 being 27+125).

From there, we can set 27/152 = x/456 in order to find the actual volume of the small cylinder from its ratio to the total volume. This gives us 81 cubic feet for the volume of the smaller cylinder.

I hope this helps! Feel free to follow up if it's still confusing or if I did a poor job explaining.