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# Solving Trig Equations- Word Problems (Grades 11-12)

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## Solving Trig Equations- Word Problems

1.
Terry often goes fishing at the end of a pier on the ocean. As the tide changes, so does the height of the water level compared to the pier. Terry finds it difficult to reel in a fish if the water level is too low compared to the pier, and thus waits till the height difference between the water level and the pier is no greater than 3 meters. The function $D(t) = 2sin(0.18 pi t) + 4$ describes the vertical distance between the water level and the pier in meters for about the next 24 hours, where $t$ is the time measured in hours since midnight. When is the first opportunity for Terry to go fishing today?
1. 6:29 am
2. 4:38 am
3. 3:40 am
4. 12:56 am
2.
A certain island is populated mainly by deer and coyotes. The coyotes mainly eat the deer, and the deer have no other predators. These two groups of animals exhibit a predator-prey relationship. There are currently a lot of deer on the island. Therefore the coyote population will increase because of abundant food. As it does, the deer population will decrease because of more predators. Then, the coyote population will decrease as their food source diminishes, which will allow the deer population to rebound, and the cycle will continue.

The deer population can be modeled by the function $D(w) = 60 cos(pi/75 w) + 160$, where $D$ is the number of deer on the island and $w$ is the number of weeks since the current date. How many weeks will it take for the deer population to decrease by 90 deer from its current level?
1. 2 weeks
2. 75 weeks
3. 25 weeks
4. 50 weeks
3.
Tim and Andrew are on a Ferris wheel. Their vertical height above the ground can be modeled by the function $H(t) = -40cos(pi/5 t) + 43$, where $H$ is measured in meters and $t$ is measured in minutes, $t=0$ corresponding to when they got on the Ferris wheel. If they are currently 56 m above the ground and are partway through their second rotation, which of the following are possibilities of how long they have been on the ride? Assume that the Ferris wheel has not stopped since Tim and Andrew got on. Choose all correct answers.
1. 3 minutes and 2 seconds
2. 16 minutes and 58 seconds
3. 13 minutes and 2 seconds
4. 11 minutes and 58 seconds
4.
During the month of February in Townsville, the temperature each day can be approximately modeled by the function $T(t) = -21.5 cos( pi/12 (t-3)) + 23.5$, where $T$ is measured in degrees Fahrenheit and $t$ is the number of hours since midnight of the last day of January. When does the temperature reach 32 °F during a typical day in February? Choose all correct answers.
1. 7:27 am
2. 10:33 am
3. 4:27 pm
4. 7:27 pm
5.
A power company that produces electricity for a certain area has found that the function $P(t) = -2200 sin(pi/12 (t+1.5)) + 7500$ closely models the average amount of power demanded throughout the day during the months of December to February, where $P$ is measured in MW and $t$ is the number of hours since the end of November. If the company wants to charge a higher rate when the power demand is approximately 95% of its maximum and above, what hours should the company charge this higher rate during these months? Round to the nearest hour.
1. From about 11:00 am to 10 pm.
2. From about 2:00 pm to 7:00 pm.
3. From about 4:00 am to 5:00 am.
4. From about 5:00 pm to 7:00 pm.
6.
A mountain resort is open all year round, but features skiing in the winter and mountain biking in the summer. The number of guests they have can be approximately modeled by the function $G(d) = 235 cos( \ 0.03443 \ (d+12) \ ) + 315$, where $G$ is the approximate number of guests on the $d^{th}$ day of the year. If the resort considers their peak season to be when they have 85% or more of their max number of guests, approximately when is their peak season? Choose all correct answers. Assume that years have 365 days.
1. Nov. 23 - Jan. 13
2. Dec. 6 - Jan. 25
3. May 25 - July 14
4. Mar. 14 - Mar. 24
7.
Kayla is currently swinging on a swing. The function $D(t) = 1.4 sin(pi/3 t) + 1.9$ describes the vertical distance she is from the ground in meters, given the amount of time that has elapsed, in seconds, from her current position $(t = 0)$. She wants to jump off the swing, but doesn't want to be too high when she does so. She plans to jump off the swing when she is about 0.8 m off the ground. Kayla also wants to be swinging forward when she jumps off, but only once she has passed the center line (where the swing is lowest). Which of the following times represent a possible opportunity for when she could jump off? Assume that she is swinging forward at $t=0$.
1. 3.8 seconds
2. 0.9 seconds
3. 11.1 seconds
4. 5.1 seconds
8.
The function $S(d) = -4.4 cos( \ 0.0172 \ (d+10) \ ) + 12.23$ gives the approximate number of hours of sunlight, $S$, for a certain city on the $d^(th)$ day of the year. Matt has a certain potted plant which he winters indoors, but keeps outside as long as there is enough light. If the plant needs at least 14 hours of sunlight each day, for about how many days each year will he be able to keep this plant outside? Assume years have 365 days.
1. 106 days
2. 135 days
3. 153 days
4. 258 days
9.
A certain cat toy is suspended from the ceiling and bounces a bird-like object up and down. The bird-like object, which is hanging on a string, easily detaches if the cat catches it. The ceiling attachment has a small motor, which keeps the bird-like object bouncing up and down as if it were on a spring. The function $H(t) = 0.5 cos(pi/3 t) + 2$ describes the height of the object from the floor, in meters, given the amount of time, in seconds, since the toy has been activated. If an average cat can jump about 1.7 m, about how long does an average cat have to catch the toy while its within range during each bouncing cycle?
1. 0.9 seconds
2. 2.1 seconds
3. 1.8 seconds
4. 4.2 seconds
10.
The town of Orson has a college located nearby. The number of apartments listed for rent exhibits a periodic trend, with the highest number of listings near the end of a term and the start of the next. The function $A(d) = 45cos(0.05163(d+4)) + 55$ represents the approximate number of apartments for rent on a given day of the year.

Grace lives in Orson, but wants to move to a new apartment. She is hoping to avoid the busiest times of the year, but also doesn't want to look during the periods when there are only a few apartments listed. If Grace wants to look for apartments when there are at least about 45 but no more than about 65 listings each day, approximately how many days each a year will she be able to look?
1. 22 days
2. 30 days
3. 60 days
4. 159 days
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