Question

A baseball card sold for $254 in 1977 and was sold again in 1990 for $408. Assume that the growth in the value V of the collector's item was exponential. a) Find the value k of the exponential growth rate. Assume Vo = 254. k= ____(Round to the nearest thousandth.) b) Find the exponential growth function in terms of t, where t is the number of years since 1977. V(t) = _____C) Estimate the value of the baseball card in 2015. (Round to the nearest dollar.)$_____d) what is the doubling time for the value of the baseball card to the nearest tenth of a year?____yearse) Find the amount of time after which the value of the baseball card will be $1655.____years

Answer 1

a)

The baseball card was sold for $254 in 1977

It was sold again in 1990 for $408

If the growth of the value (V) is exponential, you can express the said growth as:

$V(t)=a\cdot e^{kt}$

Where

a is the initial value

e is the natural number

k is the growth rate

t is the time period

Considering 1977 as the initial time t=0years, then the price the cards was sold of is the initial price:

a= $254

In 1990 the card was sold for V= $408

To determine the time period that corresponds to 1990, you have to determine the difference between both years:

$\begin{gathered} t=1990-1977 \\ t=13 \end{gathered}$

You can express the exponential growth of the card's price as:

$V(t)=254e^{kt}$

To determine the value of k, you have to replace the expression with V=408 and t=13:

$\begin{gathered} V(t)=254e^{kt} \\ 408=254e^{13k} \end{gathered}$

-First, divide both sides by 254

$\begin{gathered} \frac{408}{254}=\frac{254e^{13k}}{254} \\ \frac{204}{127}=e^{13k} \end{gathered}$

-Apply the natural logarithm to both sides of the equal sign to reverse the exponent

$\begin{gathered} \ln (\frac{204}{127})=\ln (e^{13k}) \\ \frac{204}{127}=13k \end{gathered}$

-Divide both sides of the expression by 13 to reach the value of k

$\begin{gathered} \frac{204}{127}\cdot\frac{1}{13}=\frac{13}{13}k \\ 0.1235=k \\ k\approx0.124 \end{gathered}$

b)

To determine the said equation you have to write the expression used in item a with the value of k:

$V(t)=254e^{0.124t}$

c)

To estimate the value of the baseball card in 2015, the first step is to determine the number of years passed from 1977 to 2015

$\begin{gathered} t=2015-1977 \\ t=38 \end{gathered}$

Replace the expression obtained in item b with t=38 and calculate the value of V

$\begin{gathered} V(38)=254e^{0.124\cdot38} \\ V(38)=254e^{4.712} \\ V(38)=28263.719 \\ V(38)=28263.72 \end{gathered}$

d)

To determine the doubling time indicates that you have to calculate the value of t, when the initial value will be double

$\begin{gathered} V(t)=254e^{0.124t} \\ 508=254e^{0.124t} \\ \frac{508}{254}=\frac{254e^{0.124t}}{254} \\ 2=e^{0.124t} \\ \ln (2)=\ln (e^{0.124t}) \\ \ln (2)=0.124t \\ \frac{\ln 2}{0.124}=\frac{0.124t}{0.124} \\ 5.589=t \\ t\approx5.6years \end{gathered}$

The price of the card will be double after 5.56years

e)

To determine the time it will take for the value of the card to be $1655, you cave to equal the expression with V=1655

$\begin{gathered} 1655=254e^{0.124t} \\ \frac{1655}{254}=\frac{254e^{0.124t}}{254} \\ \frac{1655}{254}=e^{0.124t} \\ \ln (\frac{1655}{254})=\ln (e^{0.124t}) \\ \ln (\frac{1655}{254})=0.124t \\ \ln (\frac{1655}{254})\div0.124=\frac{0.124}{0.124}t \\ 15.11=t \end{gathered}$

It will take 15.11years for the value of the card to be $1655.