Question

a map of a park has a scale of 1 inch to 1,000 feet. another map of the same park has a scale of 1 inch to 500 feet. which map is larger? explain or show your reasoning.

Explanation:

Step1: Understand Map Scales

A map scale of \(1\) inch to \(1000\) feet means \(1\) inch on the map represents \(1000\) feet in real life. A scale of \(1\) inch to \(500\) feet means \(1\) inch on the map represents \(500\) feet in real life.

Step2: Relate Scale to Map Size

For the same real - world park (same actual area), let's consider a certain real - world distance, say \(d = 1000\) feet.

• For the first map (scale \(1\) inch : \(1000\) feet), the length on the map to represent \(1000\) feet is \(l_1=\frac{1000}{1000}=1\) inch.
• For the second map (scale \(1\) inch : \(500\) feet), the length on the map to represent \(1000\) feet is \(l_2=\frac{1000}{500} = 2\) inches.

In general, for a given real - world distance, a smaller representative fraction (where the denominator in the scale \(1\) inch : \(x\) feet is smaller, like \(x = 500\) compared to \(x = 1000\)) will require a longer length on the map. Since the map with scale \(1\) inch to \(500\) feet uses more inches on the map to represent the same real - world distances, the map with scale \(1\) inch to \(500\) feet will be larger.

Answer:

The map with a scale of 1 inch to 500 feet is larger. Because for the same real - world distance in the park, the map with the smaller scale (1 inch represents a smaller number of real - world feet, 500 feet vs 1000 feet) will need to use more inches on the map to represent that distance, resulting in a larger - sized map.