함수와 그래프 전체 문제

[T] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle $\theta$ can be modeled by the function

$\theta ={\tan }^{-1}\frac{5.5}{x}-{\tan }^{-1}\frac{2.5}{x},$ where $x$ is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.

[T] The number of bacteria N in a culture after t days can be modeled by the function $N(t)=1300\cdot {(2)}^{t/4}.$ Find the number of bacteria present after 15 days.

[T] The concentration of hydrogen ions in a substance is denoted by $[{\text{H}}^{+}],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $\text{pH}=-\log [{\text{H}}^{+}].$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

a. Find the pH of the following substances. Round answers to one digit. b. Determine whether the substance is an acid or a base. i. Eggs: $[{\text{H}}^{+}]=1.6\,\times \,{10}^{-8}$ mol/L ii. Beer: $[{\text{H}}^{+}]=3.16\,\times \,{10}^{-3}$ mol/L iii. Tomato Juice: $[{\text{H}}^{+}]=7.94\,\times \,{10}^{-5}$ mol/L

[T] According to the World Bank, at the end of 2013 ($t=0$ ) the U.S. population was 316 million and was increasing according to the following model:

$P(t)=316{e}^{0.0074t},$ where P is measured in millions of people and t is measured in years after 2013.

a. Based on this model, what will be the population of the United States in 2020? b. Determine when the U.S. population will be twice what it is in 2013.

[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

a. Use the exponential function $Q={Q}_{0}{e}^{kt}$ to determine the value $k,$ which is the growth rate of the bacteria. Round to four decimal places. b. Determine approximately how long it takes for 200,000 bacteria to grow.

[T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.

a. Use the exponential function $P={P}_{0}{a}^{t}$ to determine the growth rate constant $a.$ Round to four decimal places. b. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.

a. Find the equation $C=f(x)$ that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

The population can be modeled by $P(t)=82.5-67.5\cos [(\pi /6)t],$ where $t$ is time in months $(t=0$ represents January 1) and $P$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as $P(t)=82.5-67.5\cos [(\pi /6)t]+t,$ where $t$ is time in months ($t=0$ represents January 1) and $P$ is population (in thousands). When is the first time the population reaches 200,000?