13. A tree casts a shadow of 17 ft. The angle of elevation from the tip of the shadow to the top of the tree is 68°. To the nearest foot, what is the height of the tree?

14. The angle of depression of a diver from the boat is 34°, and the diver is 367 ft deep. To the nearest foot, how far is the diver from the ship?

15. A 6-foot-tall person standing at a distance of 27 ft from a water tower must look up at an angle of 46° to see the top of the water tower. To the nearest foot, find the height of the water tower.

16. A hot-air balloon has risen to an altitude of 280 m. You can still see the starting point at an 18° angle of depression. To the nearest meter, how far is the balloon from the starting point?

To solve these questions, we can use trigonometry. Trigonometry involves relationships between the angles and sides of triangles. In these questions, we'll be working with right triangles, where one angle is 90 degrees.

Let's go through each question step by step:

13. To find the height of the tree, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 68 degrees, and the opposite side is the height of the tree, while the adjacent side is the length of the shadow. We can set up the equation as follows:

tan(68) = height of tree / length of shadow

Solving for the height of the tree, we get:

height of tree = tan(68) * length of shadow

Plugging in the values, we find:

height of tree = tan(68) * 17 ft

Calculating this value will give us the height of the tree to the nearest foot.

14. In this question, the angle of depression is given, and we need to find the distance from the diver to the ship. We can use the tangent function again. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 34 degrees, and the opposite side is the depth of the diver, while the adjacent side is the distance from the diver to the ship. We can set up the equation as follows:

tan(34) = depth of diver / distance from diver to ship

Solving for the distance from the diver to the ship, we get:

distance from diver to ship = depth of diver / tan(34)

Plugging in the values, we find:

distance from diver to ship = 367 ft / tan(34)

Calculating this value will give us the distance from the diver to the ship to the nearest foot.

15. In this question, we need to find the height of the water tower. We can use the tangent function once again. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 46 degrees, and the opposite side is the height of the water tower, while the adjacent side is the distance between the person and the water tower. We can set up the equation as follows:

tan(46) = height of water tower / distance between person and water tower

Solving for the height of the water tower, we get:

height of water tower = tan(46) * distance between person and water tower

Plugging in the values, we find:

height of water tower = tan(46) * 27 ft

Calculating this value will give us the height of the water tower to the nearest foot.

16. In this question, we are given the altitude of the balloon and the angle of depression. We need to find the distance the balloon is from the starting point. We can again use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 18 degrees, and the opposite side is the altitude of the balloon, while the adjacent side is the distance between the balloon and the starting point. We can set up the equation as follows:

tan(18) = altitude of balloon / distance between balloon and starting point

Solving for the distance between the balloon and the starting point, we get:

distance between balloon and starting point = altitude of balloon / tan(18)

Plugging in the values, we find:

distance between balloon and starting point = 280 m / tan(18)

Calculating this value will give us the distance between the balloon and the starting point to the nearest meter.

(13. 42ft

(14. 656 ft
(15. 34ft
(16. 906m

13. 42 ft

14. 656 ft
15. 34 ft
16. 906 m

(13)42ft

(14)656ft
(15)34ft
(16)906m