Question

On a map whose scale is $400$ miles to an inch and a half, a certain estate is represented by a rhombus having a $60^\circ$ angle. The diagonal opposite $60^\circ$ is $\dfrac3{16}$ in. The area of the estate in square miles is:

$\dfrac{2500}{\sqrt3}$

$\dfrac{1250}{\sqrt3}$

$1250$

$\dfrac{5625\sqrt3}{2}$

$1250\sqrt3$

Answer: E

Concepts:map scale rhombus equilateral triangles area

Difficulty rating: 2210

Small Hint:

In a $60^\circ$ rhombus, the shorter diagonal equals the side length

Big Hint:

The scale is $\frac{800}{3}$ miles per inch, so convert the $\frac{3}{16}$ -inch side before finding area

Solution:

The diagonal opposite the $60^\circ$ angle divides the rhombus into two equilateral triangles, so its length equals the rhombus side. The map scale is $\frac{800}{3}$ miles per inch, hence the actual side length is $\frac3{16}\cdot\frac{800}{3}=50$ miles. Therefore the rhombus area is $\begin{aligned} 50^2\sin60^\circ &=2500\cdot\frac{\sqrt3}{2}\\ &=1250\sqrt3. \end{aligned}$

Thus, the correct answer is E.

1956 AMC 12 Problem 37

Answer: E

Difficulty rating: 2210

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years