삼각함수의 그래프

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.

For the following exercises, construct functions that model the described behavior.

A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to $t,$ the month.

For the following exercises, construct functions that model the described behavior.

Daily temperatures in the desert can be very extreme. If the temperature varies from $90\text{°F}$ to $30\text{°F}$ and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the following exercises, model the described behavior and find requested values.

An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to $t,$ the number of years from now.

For the following exercises, model the described behavior and find requested values.

The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to $t,$ the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.