Question

a designer builds a model of a truck. the finished model is exactly the same shape as the original, but smaller. the scale factor is 2:9. (a) find the ratio of the volume of the model to the volume of the original. (b) find the ratio of the surface area of the model to the surface area of the original. (c) find the ratio of the width of the model to the width of the original. write these ratios in the format m:n. volume: : surface area: : width: :

Explanation:

Step1: Recall ratio - volume relationship

If the scale factor of two similar solids is \(a:b\), the ratio of their volumes is \(a^{3}:b^{3}\). Here \(a = 2\) and \(b=9\), so the ratio of the volume of the model to the volume of the original is \(2^{3}:9^{3}\).
\[2^{3}=8,9^{3}=729\]
The volume ratio is \(8:729\).

Step2: Recall ratio - surface - area relationship

If the scale factor of two similar solids is \(a:b\), the ratio of their surface - areas is \(a^{2}:b^{2}\). Here \(a = 2\) and \(b = 9\), so the ratio of the surface area of the model to the surface area of the original is \(2^{2}:9^{2}\).
\[2^{2}=4,9^{2}=81\]
The surface - area ratio is \(4:81\).

Step3: Recall ratio of corresponding lengths

The ratio of corresponding lengths (such as width) of two similar solids is equal to the scale factor. The scale factor is \(2:9\), so the ratio of the width of the model to the width of the original is \(2:9\).

Answer:

volume: \(8:729\)
surface area: \(4:81\)
width: \(2:9\)