함수와 그래프 전체 문제

[T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of $120^\circ ,$ how many inches does it move? Approximate to the nearest whole inch.

[T] The area $A$ of a square depends on the length of the side $s.$

a. Write a function $A(s)$ for the area of a square. b. Find and interpret $A(6.5).$ c. Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.

[T] The volume of a cube depends on the length of the sides $s.$

a. Write a function $V(s)$ for the volume of a cube. b. Find and interpret $V(11.8).$

[T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost $C$ to rent a car is a function of the hours $t$ the car is rented plus the flat fee.

a. Write the formula for the function that models this situation. b. Find the total cost to rent a car for 2 days and 7 hours. c. Determine how long the car was rented if the bill is $430.

[T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank.

a. Write a formula that models this situation. b. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas. c. Determine the domain and range of the function. d. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

[T] The volume V of a sphere depends on the length of its radius as $V=(4/3)\pi {r}^{3}.$ Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius $6.371\,\times \,{10}^{6}$ m.

[T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t)=6-[5/({t}^{2}+1)],$ where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

a. Express the area of the bacteria as a function of time. b. Find the exact and approximate area of the bacterial culture in 3 hours. c. Express the circumference of the bacteria as a function of time. d. Find the exact and approximate circumference of the bacteria in 3 hours.

[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost $C$ as a function of number of cupcakes made.

a. Find a linear function that relates cost C to x, the number of cupcakes made. b. Find the cost to bake 160 cupcakes. c. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)

[T] The total cost C to produce a certain item is modeled by the function $C(x)=10.50x+28,500,$ where x is the number of items produced. Determine the cost to produce 175 items.

[T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.

a. Find the linear function $y=N(t)$ that models this situation, where $N$ is the number of pages typed and t is the time in minutes. b. Use part a. to determine how many pages can be typed in 2 hours. c. Use part a. to determine how long it takes to type a 20-page assignment.

[T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $\omega ,$ and is given by $\omega =\theta /t,$ where $\theta$ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. $\theta =\frac{7\pi }{4}\text{rad},t=10$ sec b. $\theta =\frac{3\pi }{5}\text{rad},t=8$ sec c. $\theta =\frac{2\pi }{9}\text{rad},t=1$ min d. $\theta =23.76\text{rad,}t=14$ min

[T] The area of an isosceles triangle with equal sides of length x is

$\frac{1}{2}{x}^{2}\sin \theta ,$

where $\theta$ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle $\theta =5\pi /12$ rad.

[T] A particle travels in a circular path at a constant angular speed $\omega .$ The angular speed is modeled by the function $\omega =9|\cos (\pi t-\pi /12)|.$ Determine the angular speed at $t=9$ sec.

[T] An alternating current for outlets in a home has voltage given by the function

$V(t)=150\cos 368t,$

where V is the voltage in volts at time t in seconds.

a. Find the period of the function and interpret its meaning. b. Determine the number of periods that occur when 1 sec has passed.

[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function $V=f(x)=500(0.04-{x}^{2})$ for $0\le x\le 0.2.$

a. Find $x={f}^{-1}(V).$ b. Interpret what the inverse function is used for. c. Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.

[T] The cost to remove a toxin from a lake is modeled by the function

$C(p)=75p/(85-p),$ where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.

a. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake. b. Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.

[T] A race car is accelerating at a velocity given by

$v(t)=\frac{25}{4}t+54,$ where v is the velocity (in feet per second) at time t.

a. Find the velocity of the car at 10 sec. b. Find the inverse function. c. Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.

[T] The average temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

$T(x)=5+18\sin \left[\frac{\pi }{6}(x-4.6)\right],$ where $x$ is time in months and $x=1.00$ corresponds to January 1. Determine the day(s) (month and day) when the average temperature is $21^\circ \text{C}.$ Use the integer portion of your answer(s) as the month and calculate the day of the month from the decimal portion.

[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function

$D(t)=5\sin \left(\frac{\pi }{6}t-\frac{7\pi }{6}\right)+8,$ where $t$ is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.

[T] An object moving in simple harmonic motion is modeled by the function

$s(t)=-6\cos \left(\frac{\pi t}{2}\right),$ where $s$ is measured in inches and $t$ is measured in seconds. Determine the first time when the distance moved is 4.5 in.