Precalculus Quiz: Solving Trigonometric Equations In Context

The voltage in an AC circuit is modeled by $V(t)=10\sin(120\pi t)$ volts, where $t$ is in seconds. Using the model, which equation correctly represents finding the times in $0\le t\le 0.05$ when the voltage is $5$ volts?

The temperature in a greenhouse is modeled by $T(t)=10\cos\left(\frac{\pi t}{12}\right)+20$, where $t$ is hours after midnight and $0\le t<24$. Using the model, when is the temperature $15^\circ\text{C}$ during the day? Give all solutions in the interval $[0,24)$.

The temperature (in °C) during a day is modeled by $T(t)=10\cos\left(\frac{\pi t}{12}\right)+20$, where $t$ is hours after midnight. Based on the model, when is the temperature $15^\circ\text{C}$ during the first 24 hours ($0\le t<24$)?

A piston's position in an engine is modeled by x(t) = 5.5 + 4.5cos(8πt + π/4), where x is position in centimeters from a reference point and t is time in seconds.

Quality control requires measuring the piston when it reaches exactly 7.75 cm from the reference point. In a 0.5-second test cycle, how many measurement opportunities will occur, and what is the time difference between the first and last measurements?

A pendulum's angular displacement from vertical is given by θ(t) = 0.3cos(2πt + π/4), where θ is measured in radians and t is time in seconds.

A sensor triggers when the pendulum reaches an angular displacement of exactly -0.15 radians. During the first 3 seconds of motion, how many times will the sensor trigger?

The angular velocity of a wind turbine blade varies with wind speed according to ω(t) = 12 + 8sin(πt/4 + π/6), where ω is angular velocity in radians per second and t is time in seconds.

The turbine's automatic brake system engages when the angular velocity reaches exactly 18 rad/s. During a 16-second monitoring period, what is the time interval between the first and third occurrences of this angular velocity?

The temperature (in °C) over a day is modeled by $T(t)=10\cos\left(\frac{\pi t}{12}\right)+20$, where $t$ is hours after midnight. Using the model, when is the temperature $15^\circ\text{C}$ during the first 24 hours ($0\le t<24$)?

A buoy moves vertically with height (in meters) modeled by $h(t)=3\cos\left(\frac{\pi t}{4}\right)+5$, where $t$ is time in seconds. Using the model, when does the buoy first reach a height of $6.5$ m after $t=0$? (You may use technology.)

A mass on a spring has position (in cm) from equilibrium given by $y(t)=4\cos(3t)$, where $t$ is in seconds. Using the model, what is the first time after $t=0$ when the mass reaches $y=2$ cm? (Answer in radians-based seconds; you may use technology.)

A Ferris wheel rider's height above the ground is modeled by $h(t)=15\sin\left(\frac{\pi t}{20}\right)+18$, where $t$ is time in seconds after boarding. Using the model, at what time(s) in the first minute ($0\le t\le 60$) is the rider exactly $25$ meters above the ground? (You may use a graphing calculator or equation solver to approximate.)

A mass on a spring has position (in cm from equilibrium) modeled by $y(t)=4\cos(3t)$, where $t$ is in seconds. Using the model, what is the first time after $t=0$ when the mass reaches $y=2$ cm? (Give $t$ in seconds.)

A runner's distance from a starting line (in meters) is modeled by $s(t)=10\sin\left(\frac{\pi t}{5}\right)+15$, where $t$ is time in seconds. Using the model, which equation correctly represents finding the time(s) in $0\le t\le 10$ when the runner is $20$ meters from the start?

A pendulum's angular displacement is modeled by $\theta(t)=3\sin\left(\frac{\pi t}{4}\right)$ radians, where $t$ is time in seconds. Using the model, how many times in the interval $0\le t\le 16$ does the pendulum have displacement $\theta=1.5$ radians?

A mass on a spring has position (from equilibrium) modeled by $y(t)=4\cos(3t)$ cm, where $t$ is in seconds and $t\ge 0$. Based on the model, what is the first time after $t=0$ when the mass reaches $y=2$ cm? (Answer in seconds; you may use an equation solver.)

Daylight hours in a city are modeled by $D(d)=3\sin\left(\frac{2\pi(d-80)}{365}\right)+12$, where $d$ is the day of the year ($d=1$ is Jan 1). Using the model, on which day numbers $d$ in $[1,365]$ are there exactly $13.5$ hours of daylight? (Round to the nearest whole day; technology is allowed.)

A pendulum's horizontal position is modeled by $x(t)=4\cos\left(\frac{\pi t}{2}\right)$ cm, where $t$ is time in seconds and $0\le t\le 8$. Using the model, at what time(s) does the pendulum have position $x(t)=2$ cm in this interval?

A mass on a spring has position (in cm) from equilibrium given by $y(t)=4\cos(3t)$, where $t$ is time in seconds and $t\ge 0$. Using the model, what is the first time after $t=0$ when the mass reaches position $y=2$ cm? (Answer in seconds; you may use technology.)​

A Ferris wheel height (in meters) is modeled by $h(t)=10\sin\left(\frac{\pi t}{15}\right)+15$, where $t$ is time in seconds and $0\le t\le 30$. Using the model, what does the solution $t=7.5$ seconds represent if it satisfies the equation $h(t)=25$?​

The temperature (in °C) during a day is modeled by $T(t)=10\cos\left(\frac{\pi t}{12}\right)+20$, where $t$ is hours after midnight and $0\le t<24$. Based on the model, at what time(s) is the temperature $15^\circ\text{C}$?​

A rotating beacon's brightness at a sensor is modeled by $B(t)=5\cos(2t)+10$, where $t$ is in seconds and $0\le t\le 2\pi$. Using the model, solve $B(t)=15$ for $t$ in the given interval.