Question

a portion of a city map is shown in the figure at the right. if you walk along maple street between 2nd street and elm grove lane, how far do you walk?
maple street is about \\(\square\\) yd long between \\(2^{\text{nd}}\\) street and elm grove lane.
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Identify the triangle type

We have a triangle with two angles: \(47^\circ\) and \(57^\circ\), so the third angle \(C = 180^\circ - 47^\circ - 57^\circ = 76^\circ\). Let the side opposite \(47^\circ\) be \(a\), opposite \(57^\circ\) be \(b\), and opposite \(76^\circ\) be \(c = 231\) yd. We need to find the side opposite \(57^\circ\) (let's call it \(x\)) using the Law of Sines: \(\frac{x}{\sin 57^\circ}=\frac{231}{\sin 76^\circ}\).

Step2: Solve for \(x\)

Rearrange the formula: \(x=\frac{231\times\sin 57^\circ}{\sin 76^\circ}\). Calculate \(\sin 57^\circ\approx0.8387\), \(\sin 76^\circ\approx0.9703\). Then \(x = \frac{231\times0.8387}{0.9703}\approx\frac{193.7397}{0.9703}\approx199.7\).

Answer:

\(199.7\)