Chapter 6.2, Problem 18E

a.

To determine

To write: an exponential growth function that represents the weight of the catfish after t weeks during the 8-week period.

Answer to Problem 18E

  $y=0.1{\left(1.23\right)}^t$

Explanation of Solution

Given:

Initial weight of catfish = $0.1\text{ lb}$

Increase in weight = 23% each week

Concept used:

Exponential growth function:

  $y=a{\left(1+r\right)}^t$ ….. (1)

Here, the y is the final value, a is the initial value, and r is the rate of growth (in decimal form).

Calculation:

Substituting $a=0.1$ and $r=0.23$ in equation (1), the required exponential growth function is:

  $\begin{array}{c}y=a{\left(1+t\right)}^t\\ =0.1{\left(1+0.23\right)}^t\\ =0.1{\left(1.23\right)}^t\end{array}$

Conclusion:

So, the weight of the catfish after t weeks during the 8-week period is given by the function $y=0.1{\left(1.23\right)}^t$ .

b.

To determine

To determine: the weight of catfish after 4 weeks (round off to nearest thousandth).

Answer to Problem 18E

Approximately $0.229\text{ lb}$

Explanation of Solution

Given:

The weight of the catfish after t weeks during the 8-week period is given by the function: $y=0.1{\left(1.23\right)}^t$

The weight of catfish after 4 weeks is found by substituting $t=4$ in the above function.

  $\begin{array}{c}y\left(4\right)=0.1{\left(1.23\right)}^4\\ \approx 0.1\left(2.28887\right)\\ =0.228887\end{array}$

Conclusion:

So, weight of catfish after 4 weeks will be about $0.229\text{ lb}$ .