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 163​in. The area of the estate in square miles is:
Answer Choices:
A. 3​2500​
B. 3​1250​
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).