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This printable supports Common Core Mathematics Standard HSF-TF.B.5

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# Applications of Trig Functions (Grades 11-12)

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## Applications of Trig Functions

1.
Sound waves can be represented as trigonometric functions. Which of the following functions correctly describes the note A above middle C, which has a frequency of 440 Hz? 1 Hz (Hertz) is equal to one cycle per second. Assume that the note played has an amplitude of 40 dB. Also assume that if the function is graphed, the midline is the x-axis.
1. $f(t) = 40 sin(880 pi t)$
2. $f(t) = 20 sin(880 pi t) + 20$
3. $f(t) = 20 sin(440 t)$
4. $f(t) = 40 sin(440 t)$
2.
A bicycle, which is stopped at the moment, has one wheel with a slight imperfection at the top of it. The wheel is 0.6 m in diameter. If the bicycle starts moving at a steady speed of 4 m/s, which of the following functions would best model the vertical height from the ground to the imperfection on the wheel, depending on how long the bike has been in motion (in seconds)? Assume that that $t>0$.
1. $h(t) = 3/5 cos(40/3 t) + 3/10$
2. $h(t) = 3/10 cos((3pi)/5 t) + 3/10$
3. $h(t) = 3/10 cos(40/3 t) + 3/10$
4. $h(t) = 3/10 sin((3pi)/5 t) + 3/10$
3.
The following table gives the average maximum temperature each month for a small town based on historical averages. Which of the following functions best models this data, where $T$ is measured in degrees Celsius and $m$ in months? Let $m=0$ be January.

 $\ \ \ \mathbf{Month} \ \ \$ $"Jan"$ $"Feb"$ $"Mar"$ $"Apr"$ $"May"$ $"June"$ $"July"$ $"Aug"$ $"Sept"$ $"Oct"$ $"Nov"$ $"Dec"$ $\mathbf{Temp \ (°C)}$ $-15$ $-10$ $-2$ $7$ $20$ $28$ $31$ $29$ $22$ $9$ $-3$ $-10$
1. $T(m) = -23cos(pi/12 m) + 8$
2. $T(m) = 23sin(pi/6 m) + 8$
3. $T(m) = 23cos(pi/12 m) + 8$
4. $T(m) = -23cos(pi/6 m) + 8$
4.
The electricity supplied to people's homes in North America is alternating current, meaning that the current switches direction. In terms of the voltage, it oscillates between positive and negative. If the frequency of the voltage is 60 cycles per second, and the maximum voltage is 170 V (and the minimum is the negative of the maximum), which of the following functions best describes the voltage, $V$, of a typical North American outlet depending on the time, $t$, in seconds?
1. $V(t) = 170 sin(120 pi t)$
2. $V(t) = 170 sin(60 t)$
3. $V(t) = 85 sin(60 t)$
4. $V(t) = 85 sin(120 pi t) + 105$
5.
A metronome helps keep time for a musician and usually has a solid wooden base with an upside down pendulum. The way a particular metronome is currently set, the tip of of pendulum rod reaches a maximum horizontal distance of 5 cm from the center of the metronome base on each side. It takes 0.4 seconds for the pendulum to go from the center of the metronome to one of the extreme positions. If the metronome was started in a position of 2 cm to the left of the metronome's center (measured horizontally), and began moving towards the right, which of the following functions would best model the pendulum tip's horizontal distance (in centimeters) from the center of the metronome, depending on the time (in seconds) since it was started? Assume that left of the metronome base is negative.
1. $D(t) = 5 sin((5pi)/4 (t-0.105))$
2. $D(t) = 5 sin(1.6(t-0.105))$
3. $D(t) = 5 sin((5pi)/4 t)$
4. $D(t) = 5 sin((5pi)/4 (t+0.105))$
6.
A certain island has a population of deer and wolves. The deer have no other predator and the wolves mainly eat the deer. These two species exhibit a cyclical predator-prey relationship, where the deer population will grow larger and therefore provide more food for the wolves. As the wolf population grows, it begins to reduce the deer population. As their food source diminishes, the wolf population also decreases. As the wolf population decreases, the deer population will grow, and the cycle will repeat. The full cycle repeats itself once every 130 weeks. The wolf population grows to a maximum of 35 wolves and a minimum of 11 wolves. If there are currently about 23 wolves and the population is declining, approximately how many wolves will there be in 83 weeks?
1. 13 wolves
2. 32 wolves
3. 35 wolves
4. 11 wolves
7.
A circular clock is positioned on a wall such that the lowest point on the clock is 2.8 meters above the floor. The clock has a diameter of 30 cm. The minute hand of the clock reaches to the very circumference of the clock's face. The vertical distance from the floor to the tip of the minute hand can be modeled by a trigonometric function. Create a trigonometric function to model this situation, and determine about how far the minute hand is above the floor at 1:17 pm.
1. 3.10 m
2. 3.08 m
3. 2.92 m
4. 2.87 m
8.
A certain retail store has noticed that their sales increases immediately after they do a large advertising campaign on social media, but then eventually their sales start to decrease. After running 5 advertising campaigns equally spaced during the year for a few years, they notice that their sales can be modeled by a trigonometric function. They earn about $550 in sales (the maximum they earn in a single day) on the following dates: February 14, April 28, July 10, September 21, and December 3. The lowest amount they earn on a given day is$100. How much does the store typically earn on August 1? Assume all years have 365 days.
1. $429 2.$344
3. $310 4.$254
9.
The number of daylight hours varies throughout the year and can be approximately modeled by trigonometric functions. For a certain city, the maximum number of daylight hours occurs on June 21 and is 16 hours and 7 minutes. The minimum of 8 hours and 19 minutes occurs on December 21. Using this information, create a trigonometric function which models this data and determine approximately how many hours of daylight there will be on April 5. Assume that the year has 365 days.
1. 8 hours and 18 minutes
2. 11 hours and 48 minutes
3. 13 hours and 8 minutes
4. 16 hours and 8 minutes
10.
The following table gives the percentage of the moon illuminated (P.I.) on dates in August and early September for a certain city. Given that the phases of the moon are periodic and can be approximately modeled by a trigonometric function, create a function which describes the data presented in the table. Then, use this function to find what approximate percentage of the moon will be illuminated on December 1. Round the answer to the nearest percent.

 $\ \ \ \ \ \ \ \ mathbf{"Date"} \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \mathbf{"P.I."} \ \ \ \ \ \ \$ $\ \$ $\ \ \ \ \ \ \ \ mathbf{"Date"} \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \mathbf{"P.I."} \ \ \ \ \ \ \$ $\ \$ $\ \ \ \ \ \ \ \ mathbf{"Date"} \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \mathbf{"P.I."} \ \ \ \ \ \ \$ $"Aug 1"$ $96$  $"Aug 13"$ $31$  $"Aug 25"$ $40$ $"Aug 2"$ $99$  $"Aug 14"$ $22$  $"Aug 26"$ $50$ $"Aug 3"$ $100$  $"Aug 15"$ $14$  $"Aug 27"$ $60$ $"Aug 4"$ $99$  $"Aug 16"$ $8$  $"Aug 28"$ $71$ $"Aug 5"$ $96$  $"Aug 17"$ $3$  $"Aug 29"$ $80$ $"Aug 6"$ $91$  $"Aug 18"$ $0$  $"Aug 30"$ $88$ $"Aug 7"$ $85$  $"Aug 19"$ $0$  $"Aug 31"$ $94$ $"Aug 8"$ $77$  $"Aug 20"$ $3$  $"Sept 1"$ $96$ $"Aug 9"$ $69$  $"Aug 21"$ $9$  $"Sept 2"$ $99$ $"Aug 10"$ $60$  $"Aug 22"$ $16$  $"Sept 3"$ $100$ $"Aug 11"$ $50$  $"Aug 23"$ $23$  $"Sept 4"$ $99$ $"Aug 12"$ $41$  $"Aug 24"$ $29$  $"Sept 5"$ $96$
1. 3%
2. 55%
3. 84%
4. 96%
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