Download 11-7 Use Trigonometric Ratios to Solve Verbal Problems

11-7 Use Trigonometric Ratios to Solve Verbal Problems
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Date
A 32-ft telephone pole is anchored to the ground with a wire that
starts 8 ft below the top of the pole. The wire makes a 45° angle
with the ground. To the nearest foot, how long is the wire?
8 ft
x
Sketch a diagram to model the situation.
y ⫽ 32 ⫺ 8 ⫽ 24
To find y, subtract.
24
24
0.7071 艐 x
0.7071x 艐 24
24
32 ft
45°
To find x, use the sine ratio.
sin 45° ⫽ x
y
Use the ratio of the opposite side to the hypotenuse.
Use a handheld to find sin 45°.
Use cross products.
x 艐 0.7071 艐 34
Use the Division Property of Equality. Simplify.
The wire is about 34 feet long.
Solve each problem. Round the lengths to the nearest whole number.
Show your work.
1. The base of a 20-ft ladder makes a 75⬚ angle
with the ground. How far up the wall does
the ladder reach?
2. The base of a 24-ft ladder makes a 70⬚ angle
with the ground. How far up the wall does
the ladder reach?
Copyright © by William H. Sadlier, Inc. All rights reserved.
Let x ⴝ height that ladder reaches
x
x
sin 75° ⴝ 20; 0.9659 艐 20; 19 艐 x;
The ladder reaches about 19 ft.
3. Ali is standing 100 ft from the base of a tower.
Her eye level is 6 ft above the base of the tower.
If the angle of elevation from her eyes to the
top of the tower is 19⬚, what is the height
of the tower?
x
x
; 0.9397 艐 ; 23 艐 x;
24
24
about 23 ft up the wall
sin 70 ⴝ
4. Hans is standing 80 ft from the base of a
building. His eye level is 5 ft above the base of
the building. If the angle of elevation from his
eyes to the top of the building is 24⬚, what
is the height of the building?
Let b ⴝ height of the building 5 ft from the ground
b
b
tan 24 ⴝ ; 0.4452 艐 ; b 艐 36;
80
80
b ⴙ 5 艐 41
The building’s height is about 41 ft.
Let t ⴝ height of the tower 6 ft from the ground
t
t
tan 19 ⴝ
; 0.3443 艐
; t 艐 34 ft;
100
100
t ⴙ 6 艐 40
The tower’s height is about 40 ft.
5. To measure the height of a mountain, a surveyor
took two sightings, one 1000 feet farther from
the mountain than the other. The first angle of
elevation was 85⬚ and the second was 83⬚.
What is the mountain’s height?
6. To measure the height of a pole, a man took
two sightings, one 10 feet farther from the pole
than the other. The first angle of elevation
was 70⬚ and the second was 64⬚. What is the
pole’s height?
Let m ⴝ height of mountain, n ⴝ first sighting distance
m
m
tan 85 ⴝ ; m 艐 11.4301n; tan 83 ⴝ
1000 ⴙ n
n
8.1444(1000 ⴙ n) 艐 m; 8144.4 ⴙ 8.1444n ⴝ 11.4301n
8144.4 ⴝ 3.2857n; n 艐 2479; m 艐 28,335
The mountain’s height is about 28,335 ft.
Let p ⴝ pole height, q ⴝ first sighting distance
p
p
tan 70 ⴝ ; p 艐 2.7475q; tan 64 ⴝ
q
10 ⴙ q
2.0503(10 ⴙ q) 艐 p; 20.503 ⴙ 2.0503q ⴝ 2.7475q
20.503 ⴝ 0.6972q; q 艐 29.4076; p 艐 80
The pole’s height is about 80 ft.
Lesson 11-7, pages 294–297.
Chapter 11 289
For More Practice Go To:
Solve each problem. Round the lengths to the nearest whole number. Check students’ work.
Let x ⴝ distance of boat from lighthouse
25
25
tan 12 ⴝ ; 0.2126 艐 ; 0.2126x 艐 25
x
x
x 艐 118
The boat is about 118 ft away.
9. A builder is making a ramp that rises
6 ft. If the angle of elevation is 5⬚, how
long must the ramp be?
Let x ⴝ ramp length
6
sin 5 ⴝ ; 0.0872x 艐 6; x 艐 69
x
The ramp is about 69 ft long.
11. Shara is making a corner shelf shaped
like a right isosceles triangle with
a hypotenuse of 42 cm. Without using
the Pythagorean theorem, what is the
approximate length of each leg?
Use logical reasoning; Let x ⴝ measure of
90
ⴝ 45°;
each leg; each acute angle is
2
x
sin 45 ⴝ
42
x
0.7071 艐 ; x 艐 30;
42
Each leg will be about 30 cm.
8. Ruth and Dontay are in a tower
100 m above the street. Using binoculars,
they see a bike at a 9° angle of depression.
How far from the tower is the bike?
Let x ⴝ distance of bike from tower
100
100
; 0.1584 艐
; 0.1584x 艐 100
tan 9 ⴝ
x
x
x 艐 631
The bike is about 631 m away.
10. Carla is making a slide that is 460 cm long.
If the slide has a 35⬚ angle of elevation,
what will be the height?
Let x ⴝ slide height
x
; x 艐 264
460
The slide is about 264 cm high.
sin 35 ⴝ
12. Ted is setting up a tent. He connects
two 1.8-m poles at a 52° angle to make
a side of the tent. What is the length
of the base of this side?
Make a drawing; The altitude creates two 26°
x
; x 艐 0.8988(1.8)
angles; cos 26 ⴝ
1.8
2
2
x 艐 1.62; 1.8 ⴝ 1.62 ⴙ y 2; 0.6156 艐 y 2
y 艐 0.7846; 2y 艐 1.56; The base of the side
has a length of about 1.6 m.
13. Ken is near a hill. The angle of elevation from Ken’s location to the
hill’s top is 3.4°. Ken bikes towards the hill at 8 miles per hour. Thirty
minutes later, the angle of elevation from Ken’s new location is 5.1°.
What is the height of the hill in feet?
Make a drawing; Let y ⴝ height of hill, x ⴝ distance from the hill
y
At Ken’s first location: tan 3.4 ⴝ , or y 艐 0.0594x; After 30 minutes, Ken has ridden 4 mi.
x
y
From new location: tan 5.1 ⴝ
; y 艐 0.0892(x ⴚ 4); 0.0594x ⴝ 0.0892x ⴚ 0.3568
xⴚ4
ⴚ0.0298x ⴝ ⴚ0.3568; x 艐 11.9732; y 艐 0.7112 mi 艐 3755 ft; The hill has a height of about 3755 ft.
290 Chapter 11
Copyright © by William H. Sadlier, Inc. All rights reserved.
7. George is in a lighthouse 25 ft above sea
level. He looks out and sees a boat. The
angle of depression is 12°. How far
from the lighthouse is the boat?