삼각함수의 그래프

For the following exercises, find the amplitude, period, and frequency of the given function.

The displacement $h(t)$ in centimeters of a mass suspended by a spring is modeled by the function $h(t)=8\sin (6\pi t),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, find the amplitude, period, and frequency of the given function.

The displacement $h(t)$ in centimeters of a mass suspended by a spring is modeled by the function $h(t)=11\sin (12\pi t),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, find the amplitude, period, and frequency of the given function.

The displacement $h(t)$ in centimeters of a mass suspended by a spring is modeled by the function $h(t)=4\cos \left(\frac{\pi }{2}t\right),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

The displacement $h(t),$ in centimeters, of a mass suspended by a spring is modeled by the function $h(t)=-5\,\cos (60\pi t),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, $P,$ in terms of months since January, $t.$

For the following exercises, construct an equation that models the described behavior.

A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, $P,$ in terms of months since January, $t.$

For the following exercises, construct an equation that models the described behavior.

A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, $P,$ in terms of months since January, $t.$

For the following exercises, construct an equation that models the described behavior.

A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, $P,$ in terms of months since January, $t.$

For the following exercises, construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, $D,$ the end of the spring is from equilibrium in terms of seconds, $t,$ since the spring was released.

For the following exercises, construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, $D,$ the end of the spring is from equilibrium in terms of seconds, $t,$ since the spring was released.

For the following exercises, construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, $D,$ the end of the spring is from equilibrium in terms of seconds, $t,$ since the spring was released.

For the following exercises, construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, $D,$ the end of the spring is from equilibrium in terms of seconds, $t,$ since the spring was released.

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between $-0.1$ and $0.1\text{ cm,}$ effectively at rest.

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between $-0.1$ and $0.1\text{ cm,}$ effectively at rest.

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 8 times per second, was initially pulled down 32 cm from equilibrium, and the amplitude decreases by 50% each second. The second spring, oscillating 18 times per second, was initially pulled down 15 cm from equilibrium and after 4 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than $0.1\text{ cm}.$

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than $0.1\text{ cm}.$

A plane flies 1 hour at 150 mph at ${22}^{\circ }$ east of north, then continues to fly for 1.5 hours at 120 mph, this time at a bearing of ${112}^{\circ }$ east of north. Find the total distance from the starting point and the direct angle flown north of east.

A plane flies 2 hours at 200 mph at a bearing of ${60}^{\circ },$ then continues to fly for 1.5 hours at the same speed, this time at a bearing of ${150}^{\circ }.$ Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.