삼각함수의 그래프

A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function $h(t)$ gives a person’s height in meters above the ground t minutes after the wheel begins to turn.

• ⓐ Find the amplitude, midline, and period of $h(t).$
• ⓑ Find a formula for the height function $h(t).$
• ⓒ How high off the ground is a person after 5 minutes?

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y=12,000+8,000\sin (0.628x),$ where the domain is the years since 1980 and the range is the population of the city.

What is the largest and smallest population the city may have?

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y=12,000+8,000\sin (0.628x),$ where the domain is the years since 1980 and the range is the population of the city.

What are the amplitude, period, and phase shift for the function?

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y=12,000+8,000\sin (0.628x),$ where the domain is the years since 1980 and the range is the population of the city.

Over this domain, when does the population reach 18,000? 13,000?

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y=12,000+8,000\sin (0.628x),$ where the domain is the years since 1980 and the range is the population of the city.

What is the predicted population in 2007? 2010?

What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $105\text{°F}$ occurs at 5PM and the average temperature for the day is $85\text{°F}.$ Find the temperature, to the nearest degree, at 9AM.

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $84\text{°F}$ occurs at 6PM and the average temperature for the day is $70\text{°F}.$ Find the temperature, to the nearest degree, at 7AM.

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $47\text{°F}$ and $63\text{°F}$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $51\text{°F?}$

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $64\text{°F}$ and $86\text{°F}$ during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach $70\text{°F?}$

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

A Ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How much of the ride, in minutes and seconds, is spent higher than 13 meters above the ground?

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second

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, $t.$