Question

a designer builds a model of a bicycle. 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 width of the model to the width of the original. (b) find the ratio of the volume of the model to the volume of the original. (c) find the ratio of the surface area of the model to the surface area of the original. write these ratios in the format m:n. width: volume: surface area:

Explanation:

Step1: Recall ratio of linear dimensions

If the scale - factor of two similar solids is \(a:b\), the ratio of their corresponding linear dimensions (such as width) is the same as the scale - factor. Given the scale - factor is \(2:9\), for the ratio of the width of the model to the width of the original, the ratio is \(2:9\).

Step2: Recall ratio of surface areas

For two similar solids with scale - factor \(a:b\), 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 \(2^{2}:9^{2}=4:81\).

Step3: Recall ratio of volumes

For two similar solids with scale - factor \(a:b\), 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 \(2^{3}:9^{3}=8:729\).

Answer:

(a) \(2:9\)
(b) \(8:729\)
(c) \(4:81\)