use area to determine the volume and cost

Question

Two similar chocolate cakes have surface areas 144cm² and 225cm². Given that the cost of each cake is proportional to its volume. If the larger cake costs $33.75, find the cost of the smaller cake.

Answer 1

The ratio of areas is the square of the ratio of proportion. The ratio of volume is the cube of the ratio of proportion.

Ratio of areas: $$\frac{144}{225} = \left(\frac{12}{15}\right)^2$$

Therefore the ratio of proportion is $$\frac{12}{15} = {4}{5}$$

So the ratio of the volume is $$\left(\frac{4}{5}\right)^3 = \frac{64}{125}$$

Thus, the price of the small one is $$33.75·\frac{64}{125} = 17.28$$