Functions and Change: A Modeling Approach to College Algebra and Trigonometry

Bruce Crauder, Benny Evans, Alan Noell

Chapter 6 The Geometry of Right Triangles - all with Video Answers

Educators

Chapter Questions

02:34

Problem 1

A man lies 331 horizontal feet from the base of a wall. He must incline his eyes at an angle of 16.2 degrees to look at the top of the wall. How tall is the wall?

Suman Saurav Thakur

Numerade Educator

03:24

Problem 1

Find the area and perimeter of the triangle in Figure 6.19.
GRAPH CAN'T COPY.
FIGURE 6.19 Exercise 1

Calin Lupas

Numerade Educator

00:46

Problem 1

Consider an angle of $\frac{\pi}{6}$ radian with its vertex at the center of a circle of radius 5 units. What is the arc length cut by the angle? What is the area subtended by the angle?

Vinnu M

Numerade Educator

03:24

Problem 2

Find the area and perimeter of the triangle in Figure 6.20.
GRAPH CAN'T COPY.
FIGURE 6.20 Exercise 2

Calin Lupas

Numerade Educator

01:02

Problem 2

Consider an angle of 1.4 radians with its vertex at the center of a circle of radius 9 units. What is the arc length cut by the angle? What is the area subtended by the angle?

Jennifer Stoner

Numerade Educator

02:34

Problem 2

A man lies 222 horizontal feet from the base of a wall. He must incline his eyes at an angle of 6.4 degrees to look at the top of the wall. How tall is the wall?

Suman Saurav Thakur

Numerade Educator

00:59

Problem 3

Consider an angle of 25 degrees with its vertex at the center of a circle of radius 10 units. What is the arc length cut by the angle? What is the area subtended by the angle?

Uma Kumari

Numerade Educator

03:24

Problem 3

Find the area and perimeter of the triangle in Figure 6.21.
GRAPH CAN'T COPY.
FIGURE 6.21 Exercise 3

Calin Lupas

Numerade Educator

02:34

Problem 3

Calculating an angle: A man lies 270 horizontal feet from the base of a wall that is 76 feet high. At what angle must he incline his eyes in order to look at the top of the wall?

Suman Saurav Thakur

Numerade Educator

03:24

Problem 4

Find the area and perimeter of the triangle in Figure 6.22.
GRAPH CAN'T COPY.
FIGURE 6.22 Exercise 4

Calin Lupas

Numerade Educator

00:45

Problem 4

Consider an angle of $17 \mathrm{de}-$ grees with its vertex at the center of a circle of radius 3 units. What is the arc length cut by the angle? What is the area subtended by the angle?

Finian Lickona

Numerade Educator

02:34

A wall is 38 feet high. A man lies on the ground and finds that the distance from himself to the top of the wall is 83 feet. At what angle must he incline his eyes in order to look at the top of the wall?

Suman Saurav Thakur

Numerade Educator

01:44

Problem 5

A Pythagorean triple is a triple of whole numbers that form the sides of a right triangle.
a. Verify the formula
$$
\left(a^2-b^2\right)^2+(2 a b)^2=\left(a^2+b^2\right)^2 .
$$
b. Show how to use the formula from part a to make as many Pythagorean triples as you like.

Ankit Gupta

Numerade Educator

01:24

Problem 5

An angle has its vertex centered at the origin of a circle. It cuts an arc length
of 5 units and subtends an area of 9 square units. What is the radian measure of the angle? What is the radius of the circle?

Emily Burns

Numerade Educator

02:12

Problem 5

This is a continuation of Exercise 4. What is the horizontal distance from the man to the wall?

Shelby Mohamed

Numerade Educator

00:45

Problem 6

An angle has its vertex centered at the origin of a circle. It cuts an arc length of 12 units and subtends an area of 17 square units. What is the degree measure of the angle? What is the radius of the circle?

Finian Lickona

Numerade Educator

01:27

Problem 6

Verify the area formula for triangles in the case where the altitude lands outside the triangle.

Ali Soave

Numerade Educator

02:34

Problem 6

A man lies 130 horizontal feet from the base of a wall. He must incline his eyes at an angle of 13 degrees to look at the top of the wall. What is the distance from the man directly to the top of the wall?

Suman Saurav Thakur

Numerade Educator

02:34

Problem 7

A man lies 18 horizontal feet from the base of a wall. He must incline his eyes at an angle of 21 degrees to look at the top of the wall. What is the distance from the man directly to the top of the wall?

Suman Saurav Thakur

Numerade Educator

01:27

Problem 7

You and your friend are cutting pieces of a pie. Your friend cuts a piece twice as big (say, in weight) as the one you cut for yourself (see Figure 6.35). How do the central angles compare?
FIGURE CAN'T COPY.
FIGURE 6.35 Central angles

Susan Cooper

Numerade Educator

01:43

Problem 7

A triangle is isosceles if it has two sides of equal length. Suppose that an isosceles triangle has two sides of length $a$ and that the third side is the base, of length $b$. It is a fact that the altitude of such a triangle cuts the base in half. Show that the area of the triangle is given by
$$
\frac{b}{2} \sqrt{a^2-\frac{b^2}{4}} .
$$

Suman Saurav Thakur

Numerade Educator

00:38

Problem 8

This is a continuation of Exercise 7. An equilateral triangle is one in which all sides have the same length. Show that if each side has length $a$, then the area of the triangle is
$$
\frac{\sqrt{3} a^2}{4} .
$$

Amrita Bhasin

Numerade Educator

00:41

Problem 8

You have two pies, one of which has twice the diameter of the other. You cut each pie into the same number of pieces. How do the pieces of the larger pie compare (say, in weight) to those of the smaller pie?

Amrita Bhasin

Numerade Educator

01:29

Problem 8

As we noted in Section 6.1 , any triangle with sides 3,4 , and 5 is a right triangle known as the 3-4-5 right triangle.
a. Find the sine of each of the two acute angles in a 3-4-5 right triangle.
b. Use your work in part a to find the acute angles in a 3-4-5 right triangle.

Matt Just

Numerade Educator

01:06

Problem 9

If the cotangent of an acute angle $\theta$ is 5, find the degree measure of $\theta$.

Erika Bustos

Numerade Educator

02:20

Problem 9

Wichita, Kansas, is due north of Fort Worth, Texas. This means that they lie on a circle whose center is that of the Earth and whose radius equals the polar radius of the Earth, about 3950 miles (see Figure 6.36). Further, the latitude of $\mathrm{Wi}^{-}$ chita is about 37 degrees north, and that of Fort Worth is about 32 degrees north. (These angles are measured along the circle described above, starting at the equator.) How far is it from Fort Worth to Wichita?
FIGURE CAN'T COPY.
FIGURE 6.36 Distance along a meridian

Amy Jiang

Numerade Educator

01:47

Problem 9

A right triangle has legs of length $a$ and $b$.
a. Express the perimeter in terms of $a$ and $b$.
b. For the rest of this exercise, assume that the triangle has an area of 4 square units. Use this information to express the perimeter using only $a$.
c. What values of $a$ give a perimeter of 15 units?
d. What value of $a$ gives a minimum perimeter?

Kerry Thornton-Genova

Numerade Educator

07:41

Problem 10

You are facing a building that is 150 feet higher than your transit, and you must elevate your transit 20 degrees to view the top of the building. What is the distance in horizontal feet between your transit and the building?

Lourence Gonhovi

Numerade Educator

01:43

Problem 10

Area and perimeter: A right triangle has legs of length $a$ and $b$.
a. Express the perimeter in terms of $a$ and $b$.
b. For the rest of this exercise, assume that the triangle has an area of 10 square units. Use this information to express the perimeter using only $a$.
c. What values of $a$ give a perimeter of 25 units?
d. What value of $a$ gives a minimum perimeter?

Ashley High

Numerade Educator

01:37

Problem 10

This is a continuation of Exercise 9. Winnipeg, Manitoba (Canada) is due north of Fort Worth, Texas. It is about 1180 miles from Fort Worth to Winnipeg (see Figure 6.36). What is the latitude of Winnipeg?

Amy Jiang

Numerade Educator

01:18

Problem 11

You stand on the north bank of a river and look due south at a tree on the opposite bank (see Figure 6.59). Your helper on the opposite bank measures 35 yards due east to a second tree.
You must rotate your transit through an angle of 12 degrees to point toward the second tree. How wide is the river?
FIGURE CAN'T COPY.
FIGURE 6.59 Measuring distance with a transit

Prashant Bana

Numerade Educator

01:09

Problem 11

The triangles in Figure 6.37 are similar. Find the unlabeled sides.
FIGURE CAN'T COPY.
FIGURE 6.37 Exercise 11

Mitchell Cutler

Numerade Educator

01:15

Problem 11

A funny triangle? Refer to Example 6.3. Is it possible to make a right triangle with an area of 5 square units and a perimeter of 5 units? If so, how?

Taylor Shimono

Numerade Educator

01:14

Problem 12

A cannon: If you elevate a certain cannon so that it makes an angle of $t$ degrees with the ground, then the cannonball will strike the ground $\frac{m^2 \sin (2 t)}{g}$ feet downrange, where $g=32$ feet per second is acceleration due to gravity, and $m$ is the muzzle velocity in feet per second. If the muzzle velocity is 300 feet per second, what angle would you use to make the cannonball land 1000 feet downrange?

Nick Johnson

Numerade Educator

01:15

Problem 12

A funny triangle? Refer to Example 6.3. Is it possible to make a right triangle with an area of 5 square units and a perimeter of 500 units? If so, how?

Taylor Shimono

Numerade Educator

05:58

Problem 12

The triangles in Figure 6.38 are similar. Find the unlabeled sides.
FIGURE CAN'T COPY.
FIGURE 6.38 Exercise 12

Prakash Hampole

Numerade Educator

02:25

Problem 13

Dallas is 190 miles due south of Oklahoma City, and Fort Smith is 140 miles due east of Oklahoma City. An airplane flies on a direct trip from Dallas to Fort Smith.
a. What is the tangent of the angle that the flight path makes with Interstate 35 , which runs due north from Dallas to Oklahoma City?
b. What is the angle that the flight path makes with Interstate 35 ?
c. How far does the airplane fly?

Willis James

Numerade Educator

04:29

Problem 13

A right triangle has one leg of length 8 and a perimeter of 20 . How long is the other leg?

Cameron Bunney

Numerade Educator

01:09

Problem 13

The triangles in Figure 6.39 are similar. Show that $y=\frac{1}{x}$.
FIGURE CAN'T COPY.
FIGURE 6.39 Exercise 13

Mitchell Cutler

Numerade Educator

02:22

Problem 14

The triangles in Figure 6.40 are similar. Show that $y=\sqrt{x}$.
FIGURE CAN'T COPY.

Anna Jones

Numerade Educator

02:44

Problem 14

Intensity of sunlight: When incident rays of sunlight form an angle $\theta$ with a leaf, then the intensity of sunlight is reduced by a factor of $\sin \theta$, assuming that $\theta$ is between 0 and 90 degrees.
a. By what factor is the intensity reduced if the angle formed is $35^{\circ}$ ?
b. What is the angle $\theta$ if the intensity is reduced by a factor of 0.3 ?

Zachary Warner

Numerade Educator

01:00

Problem 14

A right triangle has hypotenuse of length 8 . Let $a$ and $b$ denote the lengths of the legs.
a. Use the fact that the hypotenuse is 8 to find $b$ in terms of $a$.
b. Express the perimeter in terms of $a$ alone.
c. Assume now that the perimeter is 17. Find the lengths of the legs.

Cory Kuzinski

Numerade Educator

00:44

Problem 15

A right triangle has hypotenuse of length 15 . Let $a$ and $b$ denote the lengths of the legs.
a. Use the fact that the hypotenuse is 15 to find $b$ in terms of $a$.
b. Express the perimeter in terms of $a$ alone.
c. Assume now that the perimeter is 35 . Find the lengths of the legs.

Cory Kuzinski

Numerade Educator

00:52

Problem 15

An ecologist used the accompanying diagram to estimate the diameter of a circular prey that would be optimal for the grasping claws of a praying mantid. ${ }^3$ Find a formula for the diameter $|B C|$ of the circle in terms of the length $|A C|$ and the angle $\theta$. (The vertical bars denote the length of a segment.)
GRAPH CAN'T COPY.

Christopher Stanley

Numerade Educator

01:51

Problem 15

Another type of angle measure is the grad, which is often used on foreign maps as a metric equivalent of the degree. There are 400 grads in a circle. What is the grad measure of an angle of 90 degrees? What is the grad measure of an angle of $\pi$ radians? What is the grad measure of an angle of 32 degrees?

Marshall Styczinski

Numerade Educator

01:44

Problem 16

A right triangle has hypotenuse of length 15 .
a. Explain why the sum of the lengths of the legs must be at least 15 .
b. Can the triangle have perimeter 29 ?
c. Explain why the length of each leg is at most 15 .
d. Can the triangle have perimeter 46 ?

Erika Bustos

Numerade Educator

01:11

Problem 16

The mil is a type of angle measure used in the military. The term mil is derived from milliradian, and 1000 mils make 1 radian.
a. What is the degree measure of 1 mil? Give your answer accurate to three decimal places.
b. A target is 1 yard wide and subtends an angle of $1 \mathrm{mil}$ in a soldier's field of vision. How far from the soldier is the target? Suggestion: Think of the target as an arc along the circumference of a circle.
c. How many mils are in a circle? Note: The U.S. military uses a somewhat different definition of a mil, in which there are 6400 mils in a circle. Other countries use different definitions.

Carson Merrill

Numerade Educator

00:14

Problem 16

Getting the tangent from the sine: The sine of a certain acute angle $\theta$ is $\frac{2}{3}$.
a. Make a right triangle with an acute angle $\theta$ that has a sine of $\frac{2}{3}$. Suggestion: Choose convenient lengths for the opposite side and hypotenuse.
b. Find the length of the side adjacent to $\theta$ in the right triangle you made in part a.
c. Find $\tan \theta$.

Julie Silva

Numerade Educator

02:17

Problem 17

A right triangle has hypotenuse of length 20 .
a. Let $a$ and $b$ denote the lengths of the legs. Use the fact that the hypotenuse is 20 to express $b$ in terms of $a$.
b. Express the area in terms of $a$ alone.
c. Find the lengths of the legs giving maximum area.

Willis James

Numerade Educator

01:44

Problem 17

A 10-foot vertical pole casts a 6-foot horizontal shadow (see Figure 6.41). How tall is a tree that, at the same time of day, casts a 21 -foot shadow? Suggestion: Use similar triangles.
FIGURE CAN'T COPY.

FIGURE 6.41 Similar triangles formed by shadows

Daniel Azubuike

Numerade Educator

02:19

Problem 17

The tangent of a certain acute angle $\theta$ is 5 .
a. Make a right triangle with an acute angle $\theta$ that has a tangent of 5. Suggestion: Note that $5=\frac{5}{1}$, and choose convenient lengths for the opposite and adjacent sides.
b. Find the length of the hypotenuse in the right triangle you made in part a.
c. Find $\cos \theta$.

Joshua Eastwood

Numerade Educator

00:44

Problem 18

A right triangle has hypotenuse of length 65 .
a. Let $a$ and $b$ denote the lengths of the legs. Use the fact that the hypotenuse is 65 to express $b$ in terms of $a$.
b. Express the area in terms of $a$ alone.
c. Find the lengths of the legs giving maximum area.

Luke P

Numerade Educator

01:34

Problem 18

A 20-foot ladder leans against a vertical wall, and the horizontal distance from the base of the ladder to the wall is 5 feet. One rung of the ladder is 12 feet from the base of the ladder. Use similar triangles to determine how far that rung is from the wall.

Gregory Higby

Numerade Educator

02:10

Problem 18

The cotangent of a certain acute angle $\theta$ is $\frac{2}{3}$.
a. Make a right triangle with an acute angle $\theta$ that has a cotangent of $\frac{2}{3}$. Suggestion: Choose convenient lengths for the opposite and adjacent sides.
b. Find the length of the hypotenuse in the right triangle you made in part a.
c. Find $\sin \theta$.

Ankit Gupta

Numerade Educator

01:31

Problem 19

Topographical maps show heights of mountains and depths of valleys. One difficulty in making such a map is that it is often impossible to travel over the entire terrain to be mapped. Instead, laser and radar measurements are taken.
The base of a sheer rock wall is 1.6 miles from your observation point (see Figure 6.23). Its peak is 2.2 miles away. How tall is the wall?

Prabhu Ramji

Numerade Educator

01:22

Problem 19

Consider Figure 6.42, in which an angle meets a circle. It can be shown that $\triangle A D B$ is similar to $\triangle A C E$. (The correspondence matches $\angle A D B$ with $\angle A C E$ and $\angle A B D$ with $\angle A E C$.) Show that $|A B| \times|A C|=|A D| \times|A E|$. (The vertical bars denote the length of a segment.)
FIGURE CAN'T COPY.

FIGURE 6.42 An angle and a circle

Julie Silva

Numerade Educator

01:47

Problem 19

When there is moisture on the ground, microorganisms inhabit the thin layer of water on leaves. ${ }^4$ A monolayer containing these organisms is formed on the top, and (because of the spreading pressure of this monolayer) the floating organisms spread upward when the surface is tilted. This is a dispersal method onto newly fallen leaves, for example. In the accompanying diagram, the horizontal segment represents the ground, and the angle $\theta$ measures the amount by which the surface is tilted. (We assume that $\theta$ is less than $90^{\circ}$.) The length $d$ is the distance the organisms move, and the length $v$ is the change in the elevation.
GRAPH CAN'T COPY.
a. Find a formula giving $v$ in terms of $d$ and $\theta$.
b. Assume that the distance the organisms move is 30 centimeters. Find the change in elevation if the angle $\theta$ is $15^{\circ}$ and if the angle $\theta$ is $30^{\circ}$.
c. Find the angle $\theta$ if the distance the organisms move is 30 centimeters and the change in elevation is 20 centimeters.

Nick Johnson

Numerade Educator

02:47

Problem 20

A sheer rock wall is known to be 0.16 mile high. We find that its peak is 2 miles from our observation point. How far away is the base of the rock wall?
FIGURE CAN'T COPY.
FIGURE 6.23 Radar measurement

Jacquelyn Trost

Numerade Educator

01:51

Problem 20

A circle of a certain radius has the property that if an angle has its vertex at the center of the circle, then the area it subtends is the same as the arc length. Find the radius of the circle.

Joseph Lentino

Numerade Educator

05:57

Problem 20

If an animal jumps at an angle $\theta$ to the horizontal with initial velocity $m$, then the horizontal distance $d$ that it will travel is given by
$$
d=\frac{m^2 \sin 2 \theta}{g} .
$$

Here we measure $d$ in meters and $m$ in meters per second, and $g$ is the acceleration due to gravity (about 9.8 meters per second). This model ignores air resistance.
a. Locusts typically jump at an angle $\theta=55^{\circ}$. If a locust jumps a horizontal distance of 0.8 meter, what is its initial velocity? (See Figure 6.60.)
b. Find the angle $\theta$ if the distance that an animal jumps is 1 meter and the initial velocity is 3.2 meters per second. (Assume that $\theta$ is between 0 and 45 degrees.)
FIGURE CAN'T COPY.
FIGURE 6.60 Jumping angle

Alex Garger

Numerade Educator