Chapter 7.7, Problem 39PPE

a)

To determine

To find: A function that models annual profit for a lawn-mowing business.

Answer to Problem 39PPE

Function that models the annual profit is

  $P=400{(1.05)}^t$.

Explanation of Solution

Given information: A lawn-mowing business makes a profit of $400 per year and increases 5% each year.

Formula used: Any final amount A showing growth is given by the formula,

  $A=P{\left(1+\frac{r}{100}\right)}^t$

where A is last amount, P is initial amount, r is rate of growth annually and t is time in years.

Calculation: Plug $P=400$ and $r=5$ in above formula,

  $\begin{array}{}$ A=400{\left(1+\frac{5}{100}\right)}^t=400{(1+0.05)}^t$$ \Rightarrow A=400{(1.05)}^t$\end{array}$

Conclusion: So, the function representing the annual profit A is given by the function $A=400{(1.05)}^t$ .

b)

To determine

To find: Total profit that business will earn in 10 years.

Answer to Problem 39PPE

Total profit earned in 10 years is $651.56.

Explanation of Solution

Given information: A lawn-mowing business makes a profit of $400 per year and increases 5% each year and it runs for 10 years,

Formula used: Any final amount A showing growth is given by the formula,

  $A=P{\left(1+\frac{r}{100}\right)}^t$

where A is last amount, P is initial amount, r is rate of growth annually and t is time in years.

Calculation: Plug $t=10$ in above formula,

  $\begin{array}{l}$ A=400{(1.05)}^{10}=400\times 1.6289$=651.56\end{array}$

Conclusion: So, total profit earned in 10 years is $651.56.