1 Precalculus 4.8 Applications and Models Refresh ·Solving a right triangle means to find the lengths of the sides and the measures of the angles of a right triangle. ·Some information is usually given. an angle θ and a side a, or two sides, a and b. 1 θ θ a a a θ b a b a b a

2 Precalculus 4.8 Applications and Models 2 Refresh 1. Use your unit circle to find the possible values of θ 2. Solve the missing sides of the triangle. 25° 5 x y

3 Precalculus 4.8 Applications and Models ·Solve the following right triangle. Find all sides and all angles. 3 ·a ~= 13.8 ·c ~= ·B = 55.8º

Precalculus4.8 Applications and Models Applications & Models Objectives : Use right triangles to solve real-life problems.

5 Precalculus 4.8 Applications and Models 5 Terminology Angle of elevation – angle from the horizontal upward to an object. Angle of depression – angle from the horizontal downward to an object. Horizontal Observer Angle of elevation Object Horizontal Observer Angle of depression Object

6 Precalculus 4.8 Applications and Models 6 Example A safety regulation states that the maximum angle of elevation for a rescue ladder is 72º. A fire department’s longest ladder is 110 feet. What is the maximum safe rescue height? sin 72º = h/ sin72º = h ~= h the maximum safe height is feet.

7 Precalculus 4.8 Applications and Models You try ·A ladder 22 feet long leans against the side of a house. The angle of elevation of the ladder is 70º. Find the height from the top of the ladder to the ground. ·20.67 feet 7

8 Precalculus 4.8 Applications and Models ·A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60° angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder? 8 sin θ = = Next use the inverse sine function to find . θ = sin  1 (0.875) = The angle formed by the ladder and the ground is about 61°. The painter’s plan is unsafe!

9 Precalculus 4.8 Applications and Models 9 Example – Solving Rt. Triangles At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35°, and the angle of elevation to the top of the smokestack is 53°. Find the height of the smokestack. tan(35º) = a/ tan(35º) = a (store) tan(53º) = (a + s) / tan(53º) = a + s 200tan(53º) – a = s s ~= feet

10 Precalculus 4.8 Applications and Models You try ·from a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are 35º and 47º respectively. ·draw right triangles that give a visual representation of the problem. Label the known and unknown quantities. Find the height of the steeple 10 ·tan35º = g/50 ·50tan35º = g ~= feet ·tan47º = (g+h) / 50 ·50tan47º = g+h ·50tan47º – g = h ~= 18.6

11 Precalculus 4.8 Applications and Models Swimming pool ·A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end. Find the angle of depression of the bottom of the pool. 11 ·Solution: tan(y) = 2.7 / 20 ·arctan(2.7 / 20) = y ·0.134 ~= y

12 Precalculus 4.8 Applications and Models you try ·A global positioning system satellite orbits 12,500 miles above Earth’s surface. Find the angle of depression from the satellite to the horizon. Assume the radius of the earth is 4,000 miles. ·[see #25 in the book for picture] 12

13 Precalculus 4.8 Applications and Models this is the doozy something outstanding or unique of its kind : it's gonna be a doozy of a black eye. ·In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation, to the peak, is 3.5º. After you drive 13 miles closer to the mountain, the angle of elevation is 9º. Approximate the height of the mountain. ·Do work ·Solution: ·The height of the mountain is approximately 1.3 miles high. 13

14 Precalculus 4.8 Applications and Models 14 Complete 4.7 worksheet #2. Complete 4.8 worksheet.