Question

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lesson 10:
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 \(x\) feet means \(1\) inch on the map represents \(x\) feet in real life. For a given real - world distance, the length on the map is \(\text{Map Length}=\frac{\text{Real - World Distance}}{\text{Scale Factor (feet per inch)}}\).

Step2: Compare the Two Scales

Let's consider a fixed real - world distance, say \(d\) feet.
For the first map with a scale of \(1\) inch to \(1000\) feet, the length on the map \(L_1=\frac{d}{1000}\) inches.
For the second map with a scale of \(1\) inch to \(500\) feet, the length on the map \(L_2 = \frac{d}{500}\) inches.

We know that for positive numbers \(a\), \(b\) and \(c\) (where \(b>c>0\)), \(\frac{a}{b}<\frac{a}{c}\). Here, \(a = d\) (positive real - world distance), \(b = 1000\), \(c = 500\). So, \(\frac{d}{1000}<\frac{d}{500}\), which means \(L_1 < L_2\).

Another way to think about it: A smaller scale factor (feet per inch) means that to represent the same real - world distance, we need more inches on the map. Since \(500<1000\), the map with a scale of \(1\) inch to \(500\) feet will have a larger size (because it needs to use more inches to represent the same real - world area or distance of the park)

Answer:

The map with a scale of 1 inch to 500 feet is larger.