Problem 42 A map has a scale of 288 mi to t... [FREE SOLUTION]

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Basic College Mathematics

Marvin L. Bittinger, Judith A. Beecher, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

13 Edition

Chapter 1: Problem 42

A map has a scale of 288 mi to the inch. How far apart on the map are two cities that, in reality, are 2016 mi apart? How far apart in reality are two cities that are 8 in. apart on the map?

Short Answer

Expert verified

7 inches and 2304 miles.

Step by step solution

- Understanding the scale

The scale of the map is 288 miles to the inch. This means that 1 inch on the map represents 288 miles in reality.

- Calculating map distance for given reality distance

To find the distance between two cities on the map when the real distance is 2016 miles, divide the real distance by the scale. \[ \text{Map distance} = \frac{\text{Real distance}}{\text{Scale}} = \frac{2016 \text{ mi}}{288 \text{ mi/in}} = 7 \text{ in} \]

- Calculating reality distance for given map distance

To find the real distance between two cities that are 8 inches apart on the map, multiply the map distance by the scale. \[ \text{Real distance} = \text{Map distance} \times \text{Scale} = 8 \text{ in} \times 288 \text{ mi/in} = 2304 \text{ mi} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

scale conversion

Understanding map scales is crucial in geography. The scale shows the relationship between a measurement on the map and the corresponding measurement in the real world. A scale of '288 miles to the inch' means every inch on the map equals 288 miles in reality. So, 1 inch on the map represents 288 real miles. Converting between these involves multiplication or division based on whether you鈥檙e moving from map to reality or vice versa. Always remember the core idea:

Map to reality: multiply.
Reality to map: divide.

This proportional relationship allows us to make accurate calculations.

distance calculation

Let's perform distance calculations with the provided scale. First, we need to find out how far two cities 2016 miles apart are on the map. Since we convert reality to the map, we divide the actual distance by the scale:

This tells us that two cities 2016 miles apart in real life are 7 inches apart on the map. Conversely, to find the actual distance for cities 8 inches apart on the map, we multiply the map distance by the scale:

In this way, distances can be easily calculated with the given scale.

proportional reasoning

Proportional reasoning is essential to understand scale conversions and distance calculations. It helps to think of scales as a form of ratio or fraction. In our example, the scale of 288 miles to an inch is represented by the ratio $ \frac{288 \, \text{miles}}{1 \, \text{inch}} $. Any distance on the map can be thought of proportionally to its real-life counterpart. If we increase the map distance, the real distance increases proportionally, and vice versa.

Using proportional reasoning, convert real distance to map distance or vice versa by either multiplying or dividing by the scale.
This concept plays an integral role in fields like cartography and navigation.
Always ensure the units match; converting scales typically involves maintaining the same units for accurate results.

This makes understanding and calculating scaled distances straightforward once the proportional relationship is clear.

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91影视

A map has a scale of 288 mi to the inch. How far apart on the map are two cities that, in reality, are 2016 mi apart? How far apart in reality are two cities that are 8 in. apart on the map?

Short Answer

Step by step solution

- Understanding the scale

- Calculating map distance for given reality distance

- Calculating reality distance for given map distance

Key Concepts

scale conversion

distance calculation

proportional reasoning

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