Question

mountain officials want to know the length of a new ski lift from a to c, as shown in the figure below. they measure angle dac to be 33°. they then move 1300 feet to point b and measure angle dbc to be 21°. what is the length of the new ski lift from a to c? round your answer to the nearest tenth of a foot.

Explanation:

Step1: Analyze Triangle Angles

Let \( \angle DAC = 33^\circ \), \( \angle DBC = 21^\circ \), \( AB = 1300 \) ft. Let \( AC = x \), \( BC = y \), \( CD = h \). In \( \triangle ADC \), \( \tan(33^\circ)=\frac{h}{x}\Rightarrow h = x\tan(33^\circ) \). In \( \triangle BDC \), \( \tan(21^\circ)=\frac{h}{x + 1300}\Rightarrow h=(x + 1300)\tan(21^\circ) \).

Step2: Set Equations Equal

Since \( h \) is the same, \( x\tan(33^\circ)=(x + 1300)\tan(21^\circ) \). Expand: \( x\tan(33^\circ)=x\tan(21^\circ)+1300\tan(21^\circ) \).

Step3: Solve for \( x \)

Rearrange: \( x(\tan(33^\circ)-\tan(21^\circ)) = 1300\tan(21^\circ) \). Calculate \( \tan(33^\circ)\approx0.6494 \), \( \tan(21^\circ)\approx0.3839 \). Then \( x=\frac{1300\times0.3839}{0.6494 - 0.3839} \).

Step4: Compute Value

First, \( 1300\times0.3839 = 499.07 \), \( 0.6494 - 0.3839 = 0.2655 \). Then \( x=\frac{499.07}{0.2655}\approx1879.7 \).

Answer:

The length of the new ski lift from \( A \) to \( C \) is approximately \( \boldsymbol{1879.7} \) feet.