On an original map, 1 inch in either direction represents an actual distance of 21 miles. A reproduction of the map is made so that every horizontal length is 20% shorter than on the original map, but the total printed area is 12% greater than on the original map. The horizontal and vertical dimensions are each scaled by a constant factor. On the reproduction, which choice gives the actual vertical distance, in miles, represented by 1 inch?
For a percent problem involving a change in area and changes in dimensions, convert every percent to a decimal scale factor first. Remember that an area scale factor equals the product of the horizontal and vertical scale factors. Once the missing dimension factor is found, reverse that factor when converting from a larger printed length to the actual distance represented by one inch.
Hints
Use scale factors
Write 20% shorter as a multiplication factor, and write 12% greater as a multiplication factor.
Connect area and dimensions
The area scale factor equals the horizontal scale factor multiplied by the vertical scale factor.
Interpret a larger printed dimension
If the reproduction is stretched vertically, a fixed actual distance occupies more inches, so each inch represents less actual distance.
Desmos Guide
Use the algebraic relationship
Algebra is fastest here because the area factor is the product of the two dimension factors. In Desmos, enter to calculate the vertical scale factor.
Calculate the new vertical map scale
Enter to divide the original miles-per-inch scale by the vertical enlargement factor.
Read the result
The displayed value is , so the correct choice is B.
Step-by-step Explanation
Convert the percentage changes to scale factors
A horizontal length that is 20% shorter has scale factor . An area that is 12% greater has scale factor .
Find the vertical scale factor
Let be the vertical scale factor. Because area is affected by both dimensions, . Thus, .
Relate the enlargement to the map scale
A fixed actual vertical distance takes up times as many inches on the reproduction. Therefore, the actual distance represented by each new vertical inch is the original 21 miles divided by .
Select the answer
, so the actual vertical distance represented by 1 inch is miles. The correct choice is B.