Chapter 7.8, Problem 4PPS

To determine

To write: A function to represent the value of desktop computer after years with an initial value of , then to write an equivalent function that represents the value of desktop computer with only one compounding per year, determine the effective yearly rate of the decrease in the value of desktop computer, then determine the type of computer decreasing in value at a faster rate.

Answer to Problem 4PPS

  $A(t)=450{(0.94)}^{12t}$

  $A(t)=450{(0.4759)}^t$

Laptop computer is decreasing in value at a faster rate.

Explanation of Solution

Given information: Value of laptop computer decreases at a rate of $66\%$ annually, value of desktop computer decreases at a rate of $6\%$ monthly.

Calculation:

  $A(t)=P{(1+\frac{r}{n})}^{nt}(Formulaofcompound\mathrm{int}erest)$

Substituting the values in the formula to derive the required function,

  $\begin{array}{}$ A(t)=450{(1+\frac{(-0.06)\times 12}{12})}^{12t}$$ A(t)=450{(0.94)}^{12t}$\end{array}$

Writing an equivalent function that represents the value of desktop computer with only one compounding per year,

  $\begin{array}{}$ A(t)=450{({(0.94)}^{12})}^t$$ A(t)=450{(0.4759)}^t$\end{array}$

Calculating effective yearly rate of depreciation in value of desktop computer,

  $\begin{array}{l}1+r=0.4759\\ r=-0.5241\\ r=-52.41\%\end{array}$

Effective yearly rate of depreciation in value of desktop computer is $52.41\%$ .

Yearly rate of depreciation in value of laptop computer is $66\%$ , thus, laptop computer is decreasing in value at a faster rate.