Problem: On a map whose scale is 400 miles to an inch and a half, a certain estate is represented by a rhombus having a 60∘ angle. The diagonal opposite 60∘ is 163in. The area of the estate in square miles is:
Answer Choices:
A. 32500
B. 31250
C. 1250
D. 256253
E. 12503
Solution:
The diagonal divides the rhombus into two equilateral triangles. A (rhombus) =2⋅s2⋅43=2⋅4(3/16)23. Since the map scale is 400 miles to 121 inches, 1 inch represents 32⋅400 miles and 1 square inch represents (32⋅400)2 square miles. The area of the estate is thus 162⋅42⋅323⋅328002=12503 (square miles).