Designing a zip line math problem answer

Math Questions Trigonometry

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Problem Solving Workshop: Mixed Problem Solving For use with the lessons “Use Trigonometry with Right Triangles, Define General Angles and Use Radian Measure and Evaluate Trigonometric Functions of Any Angle”

1. Multi-Step Problem The track of the Duquesne Incline in Pittsburgh, Pennsylvania is about 800 feet long at an angle of elevation of 30°. The average speed of the cable cars is about 320 feet per minute.

a. How high does the Duquesne Incline rise?

b. What is the vertical speed of the cable cars (in feet per minute)?

2. Multi-Step Problem You are designing a zip-line for a ropes course at a summer camp. A zip-line is a cable to which people can attach their safety harnesses and slide down to the ground. You want to attach one end of the cable to a pole 50 feet high and the other end to a pole 5 feet high. The maximum safe angle of elevation for the zip-line is 25°.

258

50 ft

5 ft

x

a. Find the length x of cable needed.

b. How far apart are the poles holding the cable?

3. Gridded Answer To measure the width of a river, you plant a stake on one side of the river directly across from a boulder. You then walk 70 meters to the right of the stake and measure a 67° angle between the stake and the boulder. What is the width of the river, in meters? Round your answer to the nearest whole number.

4. Open-Ended Give an example of an angle in degrees for which the cosecant is negative and the secant is positive.

5. Short Response A bicycle’s gear ratio is the number of times the freewheel turns for every one turn of the chain wheel. The table shows the number of teeth in the freewheel and chainwheel for 5 gears on a bicycle.

Gear number

Number of teeth in freewheel

Number of teeth in chainwheel

1 32 24

2 26 24

3 22 24

4 32 40

5 19 24

a. The chainwheel completes 4 rotations in third gear. Through what angle does the freewheel turn? Give your answer in both degrees and radians.

b. The chainwheel completes 3 rotations in fifth gear. Is the angle the freewheel turns more, less, or the same as the angle in part (a)? Explain your reasoning.

6. Extended Response A golf ball is hit on a level surface with an initial speed of 105 feet per second.

a. Create a table using the equation

d 5 v2

} 32

sin 2u where d is the horizontal

distance (in feet) traveled by a golf ball launched at an angle u with an initial speed v (in feet per second) for angles between 25° and 65° in 5° increments.

b. What value of u appears to maximize the horizontal distance traveled by the golf ball? Use the formula for horizontal distance traveled and the unit circle to explain why your answer makes sense.

Algebra 2 Chapter Resource Book 9-33

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Problem Solving Workshop: Mixed Problem Solving For use with the lessons “Graph Sine, Cosine, and Tangent Functions, Translate and Reflect Trigonometric Graphs, Verify Trigonometric Identities and Solve Trigonometric Equations”

1. Multi-Step Problem The depth of the ocean at a buoy reaches a maximum of 6 feet and a minimum of 2 feet. The depth d (in feet) of the ocean is modeled by

h 5 2 sin π

} 6 t 1 4 where t is the time

(in hours) with t 5 0 representing 12:00 a.m.

a. Graph the given function.

b. What is the frequency of the function? What does the frequency represent in this situation?

c. At what time(s) on the interval 0 ≤ t ≤ 12 will the depth of the ocean be 5 feet?

2. Multi-Step Problem You are standing 100 feet away from the base of the Statue of Liberty with a video camera. As you videotape the statue, you pan up the side of the statue at a rate of 5° per second.

a. Write and graph an equation that gives the height h (in feet) of the part of the statue seen through the video camera as a function of the time t (in seconds).

b. Find the change in height from t 5 0 to t 5 1, from t 5 1 to t 5 2, and from t 5 2 to t 5 3. Briefly explain what happens to h as t increases.

3. Gridded Answer Houston, Texas has a latitude of 30°N. At this latitude, the position of the sun at sunrise can be modeled by

D 5 27 sin 1 2π } 365 t 2 1.3 2 where t is the time (in days) with t 5 1 representing January 1.

In this model, D represents the number of degrees north of due east that the sun rises. What value of t corresponds to the first day that the sun is 20° north of due east at sunrise? Round your answer to the nearest integer.

4. Open-Ended Find two different sine functions that have (π, 8) as a maximum and (0, 22) as a minimum.

5. Short Response A wooden horse on a merry-go-round moves 12 inches above and 12 inches below its center position. It takes the horse 10 seconds to make 1 complete up-and-down movement.

a. Write an equation that gives the horse’s vertical position y at time t if the horse is at its center position when t 5 0.

b. Explain why you chose y 5 a sin bt or y 5 a cos bt for part (a).

6. Gridded Answer What is the amplitude of the graph below?

t

y

1

1

(0, 6)

(4, 2)

7. Extended Response You are standing 80 feet from a 300 foot building, watching as a window washer lowers himself to the ground.

window washer

you u

300 2 d

80 ft

d

a. Write an equation that gives the window washer’s distance d (in feet) from the top of the building as a function of the angle of elevation u.

b. What is the domain of the function?

c. Graph the function. Explain how the graph relates to the given situation.

Algebra 2 Chapter Resource Book 10-47