In Exercises 1-3, identify a, b, and .x. Then use them to write the exponential function that models each situation. Finally use the function to answer the question.

Question

1. When a new baby is born to the Johnsons, the family decides to invest $5000 in an account that earns 7% interest as a way to start the baby's college fund. If they do not touch that investment for 18 years, how much will there be in the college fund?

2. The local animal rescue league is trying to reduce the number of stray dogs in the county. They estimate that there are currently 400 stray dogs and that through their efforts they can place about 8% of the animals each month. How many stray dogs will remain in the county 12 months after the animal control effort has started?

3. A basket of groceries costs $96.50. Assuming an inflation rate of 1.8% per year, how much will that same basket of groceries cost in 20 years?

GPT-4o mini · Answer

To solve these problems, we'll start by identifying $ a $, $ b $, and $ x $ for each scenario and then develop the corresponding exponential function. Finally, we'll use the function to get the answer to the question posed.

Problem 1:

Identifying $ a $, $ b $, and $ x $:
- Initial amount ($ a $): $5000 (the initial investment)
- Interest rate ($ r $): 7% per year, which is 0.07 as a decimal
- Number of years ($ x $): 18 years
- The formula for compound interest is given by: \[ A = P(1 + r)^t \] Where:
  - $ A $ is the amount of money accumulated after n years, including interest.
  - $ P $ is the principal amount (the initial amount of money).
  - $ r $ is the annual interest rate (decimal).
  - $ t $ is the time the money is invested for in years.
Exponential Function: \[ A = 5000(1 + 0.07)^{18} = 5000(1.07)^{18} \]
Calculate $ A $: \[ A \approx 5000(3.439) \approx 17195.00 \] So, the college fund will have approximately $17,195.00 in 18 years.

Problem 2:

Identifying $ a $, $ b $, and $ x $:
- Initial number of stray dogs ($ a $): 400
- Rate of dogs placed each month ($ r $): 8% (0.08 as a decimal)
- The remaining stray dogs can be modeled by the formula: \[ N = a(1 - r)^t \] Where:
  - $ N $ is the number of stray dogs remaining.
  - $ a $ is the original number of stray dogs.
  - $ r $ is the rate at which the dogs are placed.
  - $ t $ is the number of months.
Exponential Function: \[ N = 400(1 - 0.08)^{12} = 400(0.92)^{12} \]
Calculate $ N $: \[ N \approx 400(0.368) \approx 147.20 \] So, approximately 147 stray dogs will remain in the county after 12 months.

Problem 3:

Identifying $ a $, $ b $, and $ x $:
- Initial cost ($ a $): $96.50
- Inflation rate ($ r $): 1.8% per year (0.018 as a decimal)
- Number of years ($ x $): 20 years
- The inflated cost can be modeled similarly: \[ C = a(1 + r)^t \]
Exponential Function: \[ C = 96.50(1 + 0.018)^{20} = 96.50(1.018)^{20} \]
Calculate $ C $: \[ C \approx 96.50(1.485) \approx 143.00 \] In 20 years, the same basket of groceries will cost approximately $143.00.