Question

example heather uses centimeter grid paper to draw a scale diagram of her garden. her real garden is 32 meters by 48 meters and heather uses a scale of 1 cm : 8 m. heather needs a smaller scale drawing, so she changes the scale to 1 cm : 16 m. now each centimeter represents 16 meters, not 8 meters. 1 calculate the dimensions of the garden using the scale 1 cm : 16 m. are the dimensions the same as they were using the scale 1 cm : 8 m? 2 how do the side lengths of heather’s new scale drawing compare to the side lengths of the original scale drawing? how can you compare the scales she used to explain this relationship?

Explanation:

Question 1

Step1: Calculate length with 1 cm:16 m

The real length is 32 meters. Using the scale \(1\space\text{cm}:16\space\text{m}\), the length on the drawing is \(\frac{32}{16}=2\space\text{cm}\). The real width is 48 meters, so the width on the drawing is \(\frac{48}{16} = 3\space\text{cm}\).

Step2: Calculate length with 1 cm:8 m

Using the scale \(1\space\text{cm}:8\space\text{m}\), the length on the drawing is \(\frac{32}{8}=4\space\text{cm}\). The width on the drawing is \(\frac{48}{8}=6\space\text{cm}\).

Step3: Compare dimensions

The dimensions with \(1\space\text{cm}:16\space\text{m}\) are \(2\space\text{cm}\times3\space\text{cm}\), and with \(1\space\text{cm}:8\space\text{m}\) are \(4\space\text{cm}\times6\space\text{cm}\). They are not the same.

Step1: Compare side lengths (new vs original scale drawing)

Original scale (\(1\space\text{cm}:8\space\text{m}\)): Length on drawing for 32 m is \(\frac{32}{8}=4\space\text{cm}\), for 48 m is \(\frac{48}{8}=6\space\text{cm}\). New scale (\(1\space\text{cm}:16\space\text{m}\)): Length on drawing for 32 m is \(\frac{32}{16}=2\space\text{cm}\), for 48 m is \(\frac{48}{16}=3\space\text{cm}\). The side lengths of the new scale drawing are half of the original scale drawing's side lengths (\(2=\frac{4}{2}\), \(3 = \frac{6}{2}\)).

Step2: Compare the scales

The original scale is \(1\space\text{cm}:8\space\text{m}\), the new scale is \(1\space\text{cm}:16\space\text{m}\). The new scale represents twice as many meters per centimeter as the original scale (\(16 = 8\times2\)). So the new scale is a reduction scale where each centimeter in the new drawing represents more real - world meters, making the drawing smaller. The relationship is that the new scale factor (meters per centimeter) is double the original scale factor, so the drawing dimensions are halved.

Answer:

The dimensions are not the same. With \(1\space\text{cm}:16\space\text{m}\), the drawing dimensions are \(2\space\text{cm}\times3\space\text{cm}\); with \(1\space\text{cm}:8\space\text{m}\), they are \(4\space\text{cm}\times6\space\text{cm}\).

Question 2