Question

1. your starting annual salary of $35,000 increases by 4% each year.
2. an item costs $4.50, and its price increases by 3.5% each year.
3. modeling real life a young channel catfish weighs about 0.1 pound. during the next 8 weeks, its weight increases by about 23% each week.

a. write a function that represents the weight of the catfish after t weeks during the 8 - week period.
b. about how much does the catfish weigh after 4 weeks?

Explanation:

Part (a)

Step1: Identify the growth model

This is a compound growth problem, so we use the exponential growth formula \( y = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate, and \( t \) is the time.
Here, the initial weight \( a = 0.1 \) pound, and the growth rate \( r = 23\% = 0.23 \).

Step2: Write the function

Substitute \( a = 0.1 \) and \( r = 0.23 \) into the formula:
\( y = 0.1(1 + 0.23)^t = 0.1(1.23)^t \)
where \( t \) is the number of weeks (with \( 0 \leq t \leq 8 \)).

Step1: Substitute \( t = 4 \) into the function

We use the function from part (a): \( y = 0.1(1.23)^t \). Substitute \( t = 4 \):
\( y = 0.1(1.23)^4 \)

Step2: Calculate \( (1.23)^4 \)

First, calculate \( 1.23^2 = 1.23 \times 1.23 = 1.5129 \)
Then, \( 1.23^4=(1.23^2)^2 = 1.5129^2 \approx 2.2889 \)

Step3: Multiply by 0.1

\( y = 0.1 \times 2.2889 \approx 0.22889 \)

Answer:

The function is \( y = 0.1(1.23)^t \)

Part (b)