Question

applications: draw a picture for each problem, then solve for the missing part. 15. the angle of elevation from a soccer ball on the ground to the top of the goal is 34°. if the goal is 8 feet tall, what is the distance from the ball to the goal? 16. sarah sights the top of the statue of liberty at an angle of elevation of 61°. if sarah is 5.5 feet tall and is standing 166 feet from the base of the statue, find its height. 17. the angle of depression from the top of a cruise ship to the top of a sailboat is 22°. sitting above water, the cruise ship is 236 feet tall while the sailboat is 27 feet tall. find the distance between the cruise ship and the sailboat. 18. a spectator in the stands spots the team mascot on the field at an angle of depression of 46°. if the spectator is sitting 35 feet above the ground, what is the horizontal distance between the spectator and the mascot?

Explanation:

Step 1: Analyze problem 15

We have a right - triangle situation with an angle of elevation of $34^{\circ}$ and the height of the goal (opposite side) $h = 8$ feet. We want to find the distance from the ball to the goal (hypotenuse). We use the sine function $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin(34^{\circ})=\frac{8}{d}$, where $d$ is the distance from the ball to the goal. Then $d=\frac{8}{\sin(34^{\circ})}$.

Step 2: Calculate $d$ for problem 15

$d=\frac{8}{\sin(34^{\circ})}\approx\frac{8}{0.5592}\approx14.3$ feet.

Step 3: Analyze problem 16

Let the height of the Statue of Liberty above Sarah's height be $h$. We have a right - triangle with an angle of elevation $\theta = 61^{\circ}$ and adjacent side $x = 166$ feet. We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So $\tan(61^{\circ})=\frac{h}{166}$, then $h = 166\times\tan(61^{\circ})$. The height of the Statue of Liberty $H=h + 5.5$.

Step 4: Calculate $H$ for problem 16

$h = 166\times\tan(61^{\circ})\approx166\times1.8040=299.464$ feet. $H=299.464 + 5.5=304.964\approx305$ feet.

Step 5: Analyze problem 17

The vertical distance between the top of the cruise - ship and the top of the sailboat is $y=236 - 27=209$ feet. The angle of depression is $22^{\circ}$, and the angle of elevation from the top of the sailboat to the top of the cruise - ship is also $22^{\circ}$. We have a right - triangle with opposite side $y = 209$ feet and we want to find the hypotenuse $d$. Using the sine function $\sin(22^{\circ})=\frac{209}{d}$, then $d=\frac{209}{\sin(22^{\circ})}$.

Step 6: Calculate $d$ for problem 17

$d=\frac{209}{\sin(22^{\circ})}\approx\frac{209}{0.3746}\approx558$ feet.

Step 7: Analyze problem 18

The angle of depression is $46^{\circ}$, so the angle of elevation from the mascot to the spectator is also $46^{\circ}$. The vertical distance (opposite side) from the mascot to the spectator is $y = 35$ feet. We want to find the horizontal distance $x$ (adjacent side). Using the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, so $\tan(46^{\circ})=\frac{35}{x}$, then $x=\frac{35}{\tan(46^{\circ})}$.

Step 8: Calculate $x$ for problem 18

$x=\frac{35}{\tan(46^{\circ})}\approx\frac{35}{1.0355}\approx34$ feet.

Answer:

1. The distance from the ball to the goal is approximately $14.3$ feet.
2. The height of the Statue of Liberty is approximately $305$ feet.
3. The distance between the cruise ship and the sailboat is approximately $558$ feet.
4. The horizontal distance between the spectator and the mascot is approximately $34$ feet.