Question

a tunnel under a river is 197.4 ft below the surface at its lowest point, as shown in the drawing. if the angle of depression of the tunnel is 5.561°, then how far apart on the surface are the entrances to the tunnel? how long is the tunnel? the entrances to the tunnel are 4055 ft apart on the surface. (round to the nearest integer as needed.) the tunnel is ft long. (round to the nearest integer as needed.)

Explanation:

Step1: Analyze right - triangle relationship

We have a right - triangle where the depth of the tunnel ($h = 197.4$ ft) is the opposite side and half of the distance between the entrances on the surface ($x$) and half of the length of the tunnel ($l$) are related to the angle of depression $\theta=5.561^{\circ}$. First, we know that $\tan\theta=\frac{h}{x}$. Since the distance between the entrances on the surface $d = 2x$. We already know $d = 4055$ ft (so $x=\frac{d}{2}$). To find the length of the tunnel $L$, we use the sine function.

Step2: Use sine function to find tunnel length

We know that $\sin\theta=\frac{h}{l}$, where $l$ is the length of half of the tunnel. So $l=\frac{h}{\sin\theta}$. Substituting $h = 197.4$ ft and $\theta = 5.561^{\circ}$, we have $\sin(5.561^{\circ})\approx0.097$. Then $l=\frac{197.4}{0.097}\approx2035.05$ ft.

Step3: Calculate total tunnel length

The length of the tunnel $L = 2l$. So $L=2\times2035.05 = 4070.1\approx4070$ ft.

Answer:

4070