Question

An engineer standing at point \(P\) wishes to determine the width of a rectangular pond. She finds distance to westernmost point \(A\) to be \(60\) m and distance to northernmost point \(B\) to be \(80\) m. If angle \(APB=60^\circ\), find \(AB\).

• A. \(13\sqrt{20}\)
• B. \(13\sqrt{10}\)
• C. \(20\sqrt{13}\)
• D. \(10\sqrt{13}\)

Hint:

Whenever two sides and included angle are given, use the cosine rule directly.

Answer 1

The Correct Option is C

Step 1: Picture the triangle.
Point $P$ is the observer, $A$ is west and $B$ is north, with $PA=60$, $PB=80$, and the angle $\angle APB=60^\circ$ between them. We want side $AB$.
Step 2: Choose the right tool.
Two sides and the included angle are known, so the Law of Cosines fits perfectly: $AB^2=PA^2+PB^2-2(PA)(PB)\cos(\angle APB)$.
Step 3: Insert the numbers.
$AB^2=60^2+80^2-2(60)(80)\cos 60^\circ$.
Step 4: Use $\cos 60^\circ=\frac12$.
$AB^2=3600+6400-2(60)(80)\cdot\frac12=3600+6400-4800$.
Step 5: Simplify.
$AB^2=10000-4800=5200$.
Step 6: Take the square root.
$AB=\sqrt{5200}=\sqrt{400\cdot13}=20\sqrt{13}$ metres.
\[ \boxed{20\sqrt{13}} \]