A map scale factor changes lengths, but an area has two length directions. First convert the scale to real kilometers per map centimeter. Then square that factor to convert the rectangle’s area, and use Desmos for the arithmetic. The tempting slip is to apply part of the length conversion only once, leaving one direction unconverted.
Hints
- Hint 1
The map measurements are in centimeters, while the scale starts in inches. Write the supplied equalities as conversion factors. Which way should each fraction face so centimeters, inches, and miles cancel in that order?
- Hint 2
A rectangle’s area is length times width. You can convert each side separately, or multiply the map area by the square of the length factor. Why would using the factor once leave one side at map size?
- Hint 3
Once the units leave square kilometers, enter the complete product in Desmos. Keep its unrounded value through the calculation; the question asks you to round only the final area.
Step-by-step
Convert the map scale, then square it
Step 1Find the length conversion
The scale links map inches to real miles. Use the given equalities as conversion factors, with each unwanted unit on the bottom of the next fraction. The inches and miles cancel, leaving kilometers per map centimeter:
- Step 2
Convert both sides of the rectangle
A rectangle’s area is length times width, so start with the two map sides. Area needs the length factor twice, once for each side. Square the whole conversion factor so both centimeters cancel:
- Step 3
Calculate and round the area
Type into Desmos. It shows about . Round the final area, not either side length: the preserve covers about square kilometers. Choice C.
Lessons that teach this
- SAT Data AnalysisIntermediateCoreConvert units with dimensional analysis
- SAT Geometry and TrigonometryIntermediateCoreUse scale factors in two and three dimensions
- SAT Geometry and TrigonometryBeginnerFind perimeter and area of plane figures
- DesmosIntermediateCoreBuild ratio, rate, and unit chains in Desmos