1Precalculus 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 θa

a

a

θ

ba

ba

ba

2Precalculus 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

3Precalculus 4.8 Applications and Models

· Solve the following right triangle. Find all sides and all angles.

3

· a ~= 13.8· c ~= 23.46· B = 55.8º

Precalculus 4.8 Applications and Models 4

4.8 Applications & Models

Objectives:Use right triangles to solve real-life problems.

5Precalculus 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

6Precalculus 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/110

• 110•sin72º = h

• 104.62 ~= h

• the maximum safe height is 104.6 feet.

7Precalculus 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

8Precalculus 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° angleof inclination. Does the painter’s plan satisfy the safetyrequirements for the use of the ladder?

8

sin θ = = 0.875

Next use the inverse sine function to find θ.

θ = sin1(0.875) = 61.044975

The angle formed by the ladder and the ground is about 61°.The painter’s plan is unsafe!

9Precalculus 4.8 Applications and Models 9

Example – Solving Rt. TrianglesAt 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/200

200•tan(35º) = a (store)

tan(53º) = (a + s) / 200

200•tan(53º) = a + s

200•tan(53º) – a = s

s ~= 125.4 feet

10Precalculus 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

· 50•tan35º = g ~= 35.01 feet

· tan47º = (g+h) / 50

· 50tan47º = g+h

· 50tan47º – g = h ~= 18.6

11Precalculus 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

12Precalculus 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]

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13Precalculus 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.

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14Precalculus 4.8 Applications and Models 14

Complete 4.7 worksheet #2.

Complete 4.8 worksheet.