Question

A designer builds a model of a minivan. The finished model is exactly the same shape as the original, but smaller. The scale factor is 3: 11.(a) Find the ratio of the surface area of the model to the surface area of the original.(b) Find the ratio of the height of the model to the height of the original.(c) Find the ratio of the volume of the model to the volume of the original.Write these ratios in the format m:n.

Answer 1

Explanation:

Recall the surface areas are measured in square units while volumes are measured in cubic units.

Hence, if the scale factor is a:b, its surface area will be a²:b² while its volume is a³:b³.

So, based on the problem, the scale factor of the model to the original size is 3:11 in which a = 3 and b = 11.

So, the surface area should be:

$\begin{gathered} a^2:b^2\Rightarrow3^2:11^2\Rightarrow9:121 \\ \end{gathered}$

a) The ratio of the surface area of the model to the original is 9:121.

For the volume, let's solve for the cube of a and b.

$a^3:b^3\Rightarrow3^3:11^3\Rightarrow27:1331$

c) The ratio of the volume of the model to the original is 27:1331.

b) The ratio of the height of the model to the original just remains constant. It is still 3:11.