Question

Karen is doing research on the Bermuda Triangle which she defines roughly by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. On her atlas Karen measures the straight-line distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton to be approximately 57 millimeters \((\mathrm{mm}), 58 \mathrm{~mm}\) and \(53.5 \mathrm{~mm},\) respectively. If the actual distance from Fort Lauderdale to San Juan is 1046 miles, approximate the actual distances from San Juan to Hamilton and from Hamilton to Fort Lauderdale.

Short answer

The actual distance from San Juan to Hamilton is approximately 964.819 miles, and from Hamilton to Fort Lauderdale is approximately 1028.938 miles.

Step-by-step solution

Understanding the Ratios

First, find the map-to-reality ratio using the given actual distance between Fort Lauderdale and San Juan. The measured distance on the map is 58 mm and the corresponding real-world distance is 1046 miles. So, the ratio is \( \text{ratio} = \frac{1046 \text{ miles}}{58 \text{ mm}} = 18.034 \text{ miles/mm} \)

Calculating Distance from San Juan to Hamilton

To find the actual distance between San Juan and Hamilton, multiply the map distance by the ratio. The map distance from San Juan to Hamilton is 53.5 mm. So, \( \text{Actual distance} = 53.5 \text{ mm} \times 18.034 \text{ miles/mm} = 964.819 \text{ miles} \)

Calculating Distance from Hamilton to Fort Lauderdale

Similarly, for the distance from Hamilton to Fort Lauderdale, use the map distance 57 mm. The actual distance is \( \text{Actual distance} = 57 \text{ mm} \times 18.034 \text{ miles/mm} = 1028.938 \text{ miles} \)

Key concepts

map-to-reality ratio

When interpreting distances on a map, it's crucial to understand the concept of the map-to-reality ratio.
This ratio tells us how much real-world distance corresponds to a certain measurement on the map. For Karen's exercise, the map-to-reality ratio is calculated using the known map and real-world distances between Fort Lauderdale and San Juan.
She measured this distance as 58 mm on the map, which is 1046 miles in the real world. Hence, the ratio, referred to also as scale, is found by dividing the real distance by the map distance.
\[ \text{ratio} = \frac{1046 \text{ miles}}{58 \text{ mm}} = 18.034 \text{ miles/mm} \] This ratio indicates that each millimeter on the map represents 18.034 miles in reality.

proportionality

Proportionality in map measurements ensures that the relationship between distances on the map and their real-world counterparts is consistent.
Given the map-to-reality ratio, we can use proportionality to find unknown real-world distances from known map distances. For instance, if we know the distance on the map between San Juan and Hamilton is 53.5 mm, we can calculate the actual distance using the established ratio:
\[ \text{Actual distance} = 53.5 \text{ mm} \times 18.034 \text{ miles/mm} = 964.819 \text{ miles} \] Thus, proportional calculations are fundamental in mapping exercises, ensuring accurate real-world distance estimations.

Bermuda Triangle distances

The Bermuda Triangle is a frequently studied and somewhat mysterious area defined by three points: Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida.
Karen's research involves measuring the linear distances between these points on a map to estimate the actual distances. Here are her findings:

• Hamilton to Fort Lauderdale: 57 mm on the map
• Fort Lauderdale to San Juan: 58 mm on the map
• San Juan to Hamilton: 53.5 mm on the map

By applying the map-to-reality ratio, Karen can estimate the actual real-world distances, as demonstrated in the provided step-by-step solution. This methodology helps in approximating distances even when not directly measurable.

real-world distance estimation

Estimating real-world distances from map measurements is a practical skill applicable in various fields such as geography, navigation, and research.
In Karen's case, after determining the map-to-reality ratio, she calculates actual distances for other segments of the Bermuda Triangle. Here's how she did it for two measures:

• From San Juan to Hamilton: \[ \text{Actual distance} = 53.5 \text{ mm} \times 18.034 \text{ miles/mm} = 964.819 \text{ miles} \]


• From Hamilton to Fort Lauderdale: \[ \text{Actual distance} = 57 \text{ mm} \times 18.034 \text{ miles/mm} = 1028.938 \text{ miles} \]

These calculations showcase how map distances, coupled with a determined ratio, can help in efficiently estimating large real-world distances. This method is valuable when direct measurement isn't feasible, enhancing our ability to interpret mapped data accurately.