Question

1. a surfer is riding a 7 - foot wave. the angle of depression from the surfer to the shoreline is 10°. what is the distance from the surfer to the shoreline? 14. a cell phone tower is anchored by two cables on each side for support. the cables stretch from the top of the tower to the ground, with each being equidistant from the base of the tower. the angle of depression from the top of the tower to the point in which the cable reaches the ground is 23°. if the tower is 140 feet tall, find the ground distance between the cables.

Explanation:

Step1: Solve Q13: Define right triangle

Let $h = 7$ ft (wave height, opposite side), $\theta = 10^\circ$ (angle of depression, equal to angle of elevation from shore to surfer). Let $d$ = distance (adjacent side).

Step2: Q13: Use tangent function

$\tan(\theta) = \frac{h}{d}$
Rearrange to solve for $d$:
$d = \frac{h}{\tan(\theta)}$
Substitute values:
$d = \frac{7}{\tan(10^\circ)}$
Calculate $\tan(10^\circ) \approx 0.1763$, so $d \approx \frac{7}{0.1763} \approx 39.7$ ft

Step3: Solve Q14: Find single cable ground distance

Tower height $H = 140$ ft, angle of depression $23^\circ$ (equal to angle of elevation from ground anchor to tower top). Let $x$ = distance from tower base to one anchor.
$\tan(23^\circ) = \frac{H}{x}$
Rearrange:
$x = \frac{H}{\tan(23^\circ)}$
Calculate $\tan(23^\circ) \approx 0.4245$, so $x \approx \frac{140}{0.4245} \approx 329.8$ ft

Step4: Q14: Total distance between cables

Since anchors are equidistant on both sides, total distance $D = 2x$
$D \approx 2 \times 329.8 = 659.6$ ft

Answer:

1. The distance from the surfer to the shoreline is approximately 39.7 feet.
2. The ground distance between the cables is approximately 659.6 feet.