Question

mountain officials want to build a new ski lift from b to c, as shown in the figure below. the distance from a to c is 1480 feet. they measure angle dac to be 36° and angle dbc to be 25°. what is the distance from a to b? round your answer to the nearest tenth of a foot.

Explanation:

Step1: Find the length of \(CD\)

In right - triangle \(ADC\), \(\sin\angle DAC=\frac{CD}{AC}\).
Given \(AC = 1480\) feet and \(\angle DAC=36^{\circ}\), then \(CD = AC\times\sin\angle DAC\).
\(CD=1480\times\sin36^{\circ}\approx1480\times0.5878 = 869.944\) feet.

Step2: Find the length of \(BD\)

In right - triangle \(BDC\), \(\tan\angle DBC=\frac{CD}{BD}\).
Given \(\angle DBC = 25^{\circ}\) and \(CD\approx869.944\) feet, then \(BD=\frac{CD}{\tan\angle DBC}\).
\(\tan25^{\circ}\approx0.4663\), so \(BD=\frac{869.944}{0.4663}\approx1865.6\) feet.

Step3: Find the length of \(AD\)

In right - triangle \(ADC\), \(\cos\angle DAC=\frac{AD}{AC}\).
Given \(AC = 1480\) feet and \(\angle DAC = 36^{\circ}\), then \(AD=AC\times\cos\angle DAC\).
\(\cos36^{\circ}\approx0.8090\), so \(AD = 1480\times0.8090=1197.32\) feet.

Step4: Calculate the length of \(AB\)

Since \(AB=BD - AD\).
\(AB=1865.6-1197.32 = 668.3\) feet.

Answer:

\(668.3\) feet