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 3:20. (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. surface area: height: volume:

Explanation:

Step1: Recall scale - factor rules for similar - shaped objects

If the scale factor of two similar - shaped objects is \(k\), the ratio of their corresponding linear dimensions (such as height) is equal to the scale factor, the ratio of their surface areas is \(k^{2}\), and the ratio of their volumes is \(k^{3}\). Here, the scale factor \(k=\frac{3}{20}\).

Step2: Find the ratio of the surface area

The ratio of the surface area of the model to the surface area of the original is \(k^{2}\). Substitute \(k = \frac{3}{20}\) into the formula for the ratio of surface areas. \(k^{2}=(\frac{3}{20})^{2}=\frac{9}{400}\), so the ratio is \(9:400\).

Step3: Find the ratio of the height

The ratio of the height of the model to the height of the original is equal to the scale factor. Since \(k=\frac{3}{20}\), the ratio is \(3:20\).

Step4: Find the ratio of the volume

The ratio of the volume of the model to the volume of the original is \(k^{3}\). Substitute \(k=\frac{3}{20}\) into the formula for the ratio of volumes. \(k^{3}=(\frac{3}{20})^{3}=\frac{27}{8000}\), so the ratio is \(27:8000\).

Answer:

(a) \(9:400\)
(b) \(3:20\)
(c) \(27:8000\)