Question

1. jada is standing 10 feet from the base of a tree and spots a nest sitting on a branch. the angle of elevation from the ground where she is standing to the nest is 55°. find the height of the nest.
2. the angle of elevation from the top of a 95 - foot tall building to a hot air balloon in the sky is 76°. if the horizontal distance between the building and the hot air balloon is 354 feet, find the height of the hot air balloon.
3. a fire hydrant sits 72 feet from the base of a 125 - foot tall building. find the angle of elevation from the fire hydrant to the top of the building.
4. a surfer is riding a 7 - foot wave. the angle of depression from the surfer to the shoreline is 10°. what is the horizontal distance from the surfer to the shoreline?
5. 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: Recall tangent - angle relationship

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Solve problem 10

Let the height of the nest be $h$. We know the adjacent side to the angle of elevation $\theta = 55^{\circ}$ is $x = 10$ feet. Using $\tan\theta=\frac{h}{x}$, we have $\tan(55^{\circ})=\frac{h}{10}$. So, $h = 10\times\tan(55^{\circ})\approx10\times1.4281 = 14.281$ feet.

Step3: Solve problem 11

Let the height of the hot - air balloon above the building be $h$. The adjacent side to the angle of elevation $\theta = 76^{\circ}$ is $x = 354$ feet. Using $\tan\theta=\frac{h}{x}$, we get $h = 354\times\tan(76^{\circ})$. Since $\tan(76^{\circ})\approx4.0108$, then $h\approx354\times4.0108=1429.8232$ feet. The height of the hot - air balloon is $H=h + 95=1429.8232+95 = 1524.8232$ feet.

Step4: Solve problem 12

Let the angle of elevation be $\theta$. The opposite side is $y = 125$ feet and the adjacent side is $x = 72$ feet. Using $\tan\theta=\frac{y}{x}$, we have $\tan\theta=\frac{125}{72}\approx1.7361$. Then $\theta=\arctan(1.7361)\approx60^{\circ}$.

Step5: Solve problem 13

The angle of depression from the surfer to the shoreline is $10^{\circ}$. The height (opposite side) is $y = 7$ feet. Let the horizontal distance (adjacent side) be $x$. Since the angle of depression is equal to the angle of elevation from the shoreline to the surfer, and $\tan\theta=\frac{y}{x}$, we have $\tan(10^{\circ})=\frac{7}{x}$. So, $x=\frac{7}{\tan(10^{\circ})}\approx\frac{7}{0.1763}=39.7164$ feet.

Step6: Solve problem 14

The angle of depression is $23^{\circ}$, and the height of the tower (opposite side) is $y = 140$ feet. Let the distance from the base of the tower to the point where the cable reaches the ground be $x$. Since the angle of depression is equal to the angle of elevation from the ground - point to the top of the tower, and $\tan\theta=\frac{y}{x}$, we have $\tan(23^{\circ})=\frac{140}{x}$. So, $x=\frac{140}{\tan(23^{\circ})}\approx\frac{140}{0.4245}=329.80$. Since there are two cables on each side and they are equidistant from the base of the tower, the distance between the cables is $2x\approx659.6$ feet.

Answer:

1. The height of the nest is approximately $14.28$ feet.
2. The height of the hot - air balloon is approximately $1524.82$ feet.
3. The angle of elevation is approximately $60^{\circ}$.
4. The horizontal distance from the surfer to the shoreline is approximately $39.72$ feet.
5. The ground distance between the cables is approximately $659.6$ feet.