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} (408)/(254)=(254e^(13k))/(254) \\ (204)/(127)=e^(13k) \end{gathered}`

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

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

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

`\begin{gathered} (204)/(127)\cdot(1)/(13)=(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) \\ (508)/(254)=(254e^(0.124t))/(254) \\ 2=e^(0.124t) \\ \ln (2)=\ln (e^(0.124t)) \\ \ln (2)=0.124t \\ (\ln 2)/(0.124)=(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) \\ (1655)/(254)=(254e^(0.124t))/(254) \\ (1655)/(254)=e^(0.124t) \\ \ln ((1655)/(254))=\ln (e^(0.124t)) \\ \ln ((1655)/(254))=0.124t \\ \ln ((1655)/(254))/0.124=(0.124)/(0.124)t \\ 15.11=t \end{gathered}`

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