Question

1. a map of a rectangular park has a length of 4 inches (in.) and a width of 6 in. it uses a scale of 1 in for every 30 miles (mi.). a. what is the actual area of the park? show how you know. b. the map needs to be reproduced at a different scale so that it has an area of 6 square inches (sq. in) and can fit in a brochure. at what scale should the map be reproduced so that it fits on the brochure? show your reasoning. 4. noah drew a scaled copy of polygon p and labeled it polygon q. if the area of polygon p is 5 square units (sq. units), what scale factor did noah apply to polygon p to create polygon q? explain or show how you know.

Explanation:

Step1: Find actual length

Given scale is 1 in for 30 miles. Map - length is 4 in. So actual length $l = 4\times30=120$ miles.

Step2: Find actual width

Map - width is 6 in. So actual width $w = 6\times30 = 180$ miles.

Step3: Calculate actual area

Area of rectangle $A=l\times w$. Substituting values, $A = 120\times180=21600$ square miles.

Step4: For part B, find original map area

Original map area $A_{map1}=4\times6 = 24$ square inches.

Step5: Let the scale - factor for new map be $k$.

The ratio of the areas of two similar figures is equal to the square of the scale - factor. Let the new map area $A_{map2}=6$ square inches. Then $\frac{A_{map2}}{A_{map1}}=k^{2}$. Substituting values, $\frac{6}{24}=k^{2}$, so $k^{2}=\frac{1}{4}$, and $k=\frac{1}{2}$.

Step6: Find new scale

Since the original scale is 1 in for 30 miles, the new scale is 1 in for $30\times2 = 60$ miles.

Answer:

A. 21600 square miles
B. 1 in for 60 miles