Question

Write an exponential function to model each situation. Find each amount after the specified time.

14. A population of 1,236,000 grows 1.3% per year for 10 years.

15. A population of 752,000 decreases 1.4% per year for 18 years.

16. A new car that sells for $18,000 depreciates 25% each year for 4 years.

Answer 1

To model population growths or decays and car depreciation with exponential functions, we use P(t) = P_0(1 + r)^t. For each given scenario, an exponential function is developed using the initial value, growth rate, and time period, allowing us to find the expected values after the given time frames.

Step-by-step explanation:

To write exponential functions to model each situation and find the amount after the specified time, we use the formula P(t) = P_0(1 + r)^t, where P(t) is the population at time t, P_0 is the initial population, r is the growth (or decay) rate, and t is the time in years.

For the problems:

A population of 1,236,000 grows 1.3% per year for 10 years. The exponential function is P(t) = 1,236,000(1 + 0.013)^10.

A population of 752,000 decreases 1.4% per year for 18 years. The exponential function is P(t) = 752,000(1 - 0.014)^18.

A new car that sells for $18,000 depreciates 25% each year for 4 years. The exponential function is P(t) = 18,000(1 - 0.25)^4.

Answer 2

14. X = 1236000(1.013^t)

1,406,413.169 at t = 10

15. X = 752000 × (0.86^t)

49,795.48457 at t = 18

16. X = 18000 × 0.75^t

5,695.3125 at t = 4

Step-by-step explanation:

14.

X = 1236000 × (1 + 1.3%)^t

X = 1236000(1.013^t)

At t = 10,

X = 1236000 × 1.013¹⁰

= 1406413.169

15. 752000 × (1 - 1.4%)^t

X = 752000 × (0.86^t)

At t = 18

X = 752000 + 0.86¹⁸

= 49795.48457

16.

X = 18000 × (1 - 25%)^t

X = 18000 × 0.75^t

At t = 4

X = 18000 × 0.75⁴

X = 5695.3125