指数增长・衰减与复利

The population P of a certain city y years after the last census is modeled by the equation below, where r is a constant and P₀ is the population when y = 0.
P = P₀(1+ r )^y
If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?

\( f\left(x\right)=5,470\left(0.64\right)^{\frac{x}{12}} \) 

The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p% of its value the preceding year, what is the value of p ?

\( P\left(t\right)=290\left(1.04\right)^{\left(\frac{4}{6}\right)t} \) 

The function \( P \) models the population, in thousands, of a certain city t years after 2005. According to the model, the population is predicted to increase by n% every 18 months. What is the value of n ?

For the function f, f (0) = 86, and for each increase in x by 1, the value of f(x) decreases by 80% . What is the value of f (2)?

Kao measured the temperature of a cup of hot chocolate placed in a room with a constant temperature of 70 degrees Fahrenheit (°F). The temperature of the hot chocolate was 185°F at 6:00 p.m. when it started cooling. The temperature of the hot chocolate was 156°F at 6:05 p.m. and 135°F at 6:10 p.m. The hot chocolate’s temperature continued to decrease. Of the following functions, which best models the temperature T(m) , in degrees Fahrenheit, of Kao’s hot chocolate m minutes after it started cooling?

p(t) =90,000(1.06)^t

The given function p models the population of Lowell t years after a census. Which of the following functions best models the population of Lowell m months after the census?

The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ?

The first term of a sequence is 9. Each term after the first is 4 times the preceding term. If w represents the n th term of the sequence, which equation gives w in terms of n ?

Two variables, x and y, are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 4. When x =0, y =200. Which equation represents this relationship?

A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per milliliter reach 5.12\( \times \)10⁸?

A sample of a certain isotope takes 29 years to decay to half its original mass. The function \( s(t)=184(0.5)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 184-gram sample starts to decay. Which statement is the best interpretation of \( s(87)=23 \) in this context?

A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function ℎ estimates that the maximum height reached after each time the ball hits the ground is 85% of the maximum height reached after the previous time the ball hit the ground. Which equation defines ℎ, where ℎ(n) is the estimated maximum height of the ball after it has hit the ground n times and n is a whole number greater than 1 and less than 10?