Chapter 7.8, Problem 2CYU

To determine

To write: A function to represent the value of mitt after $t$ years when initial value $=\$650$ , the write an equivalent function that represents the value of mitt with only one compounding per year, determine the effective yearly rate of growth of value mitt, and then at last determine her collectible which is growing at faster rate.

Answer to Problem 2CYU

  $A(t)=650{(1.0057)}^{12t}$

  $A(t)=650{(1.07058561)}^t$

Value of baseball card is increasing at a faster rate.

Explanation of Solution

Given information: Value of rare baseball card increases by $8.1\%$ every year while value of mitt increases by $0.57\%$ each month.

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)=650{(1+\frac{0.0057\times 12}{12})}^{12t}$$ A(t)=650{(1.0057)}^{12t}$\end{array}$

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

  $\begin{array}{}$ A(t)=650{({(1.0057)}^{12})}^t$$ A(t)=650{(1.07058561)}^t$\end{array}$

Calculating effective yearly rate of growth of mitt’s value,

  $\begin{array}{l}1+r=1.07058561\\ \\ r=0.07058561\\ \\ r=7.058561\%\end{array}$

Effective yearly rate of growth of mitt’s value is $7.058561\%$ .

Value of baseball card is increasing at a rate of $8.1\%$ per year, thus, value of baseball card is increasing at a faster rate.