A topographic map uses a scale of \frac{3}{8}\text{ inch} = 15\text{ miles}. A rectangular… — Armed Services Vocational Aptitude Battery (ASVAB)

A topographic map uses a scale of $\frac{3}{8}\text{ inch} = 15\text{ miles}$ . A rectangular section of a national forest is shown on this map with dimensions of $1\frac{1}{2}\text{ inches}$ by $2\frac{1}{4}\text{ inches}$ . If a conservation program plans to plant $400$ trees per square mile throughout this entire section, how many thousands of trees will be planted in total?

Answer: 2160 thousands

Answer

2160

The correct answer is found by first determining the unit scale factor. Since 3/8 inch represents 15 miles, dividing 15 by 3/8 reveals that 1 inch represents 40 miles. Multiplying the map dimensions (1.5 inches and 2.25 inches) by 40 gives the actual dimensions of 60 miles and 90 miles. The actual area is the product of these dimensions, or 5,400 square miles. Multiplying the area by 400 trees per square mile yields 2,160,000 total trees. Finally, dividing by 1,000 gives the answer in thousands: 2,160.

Step-by-Step Solution

Calculate the unit scale (miles per inch).

\frac{3}{8}\text{ inch} = 15\text{ miles} \Rightarrow 1\text{ inch} = 15 \times \frac{8}{3} = 40\text{ miles}

Finding the value of 1 inch makes it much easier to convert the mixed-fraction map dimensions.

Convert the map width to actual miles.

1\frac{1}{2}\text{ inches} = 1.5 \times 40 = 60\text{ miles}

The linear scale applies directly to one-dimensional measurements.

Convert the map length to actual miles.

2\frac{1}{4}\text{ inches} = 2.25 \times 40 = 90\text{ miles}

Applying the same linear scale factor to the other dimension.

Calculate the actual area of the forest.

60\text{ miles} \times 90\text{ miles} = 5400\text{ square miles}

The total planting area is required to determine how many trees are needed.

Calculate the total number of trees.

5400\text{ square miles} \times 400\text{ trees/square mile} = 2,160,000\text{ trees}

Applying the rate of 400 trees per square mile yields the absolute total.

Convert the final answer to the requested unit (thousands).

2,160,000 \div 1000 = 2160\text{ thousands}

The question specifically asks 'how many thousands of trees'.

Key Concept

Converting mixed-fraction map scales into actual dimensions to calculate applied area.

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