Answer:
41. 49 years
42. 34 years and 12 days
Step-by-step explanation:
As the account increases by a constant percentage each year, we can use the exponential growth formula to write a function to model the situation.
[tex]\boxed{\begin{minipage}{8 cm}\underline{Exponential Growth Formula}\\\\$ f(t)=a\left(1+r\right)^{t}$\\\\where:\\\\ \phantom{ww}$\bullet$ $f(t)$ is the account balance. \\ \phantom{ww}$\bullet$ $a$ is the principal amount.\\ \phantom{ww}$\bullet$ $r$ is the interest rate (in decimal form). \\ \phantom{ww}$\bullet$ $t $ is the time (in years). \\ \end{minipage}}[/tex]
Given values:
- f(t) = $15,000
- a = $5,000
- r = 2.3% = 0.023
Substitute the given values into the formula, and solve for t:
[tex]\begin{aligned}f(t)&=a(1+r)^t\\\\\implies 15000&=5000(1+0.023)^t\\15000&=5000(1.023)^t\\3&=(1.023)^t\\\\\textsf{Take natural logs:} \quad \ln (3)&=\ln (1.023)^t\\\ln (3)&=t\ln (1.023)\\t&=\dfrac{\ln (3)}{\ln (1.023)}\\t&=48.3129760...\end{aligned}[/tex]
The account balance will reach $15,000 during the 48th year. As the interest is applied annually, we need to round up to the nearest whole year. Therefore, the account balance will reach $15,000 after 49 years.
Note: At the end of 48 years, the account balance will be $14,893.63, and after 49 years it will be $15,236.18.
[tex]\hrulefill[/tex]
As the baseball's value increases by a constant percentage each year, we can use the exponential growth formula to write a function to model the situation.
[tex]\boxed{\begin{minipage}{8 cm}\underline{Exponential Growth Formula}\\\\$ V(t)=a\left(1+r\right)^{t}$\\\\where:\\\\ \phantom{ww}$\bullet$ $V(t)$ is the value of the baseball card. \\ \phantom{ww}$\bullet$ $a$ is the initial value of the card in 1980.\\ \phantom{ww}$\bullet$ $r$ is the interest rate (in decimal form). \\ \phantom{ww}$\bullet$ $t $ is the time (in years). \\ \end{minipage}}[/tex]
Given values:
- V(t) = $2,000
- a = $200
- r = 7% = 0.07
Substitute the given values into the formula, and solve for t:
[tex]\begin{aligned}V(t)&=a(1+r)^t\\\\\implies 2000&=200(1+0.07)^t\\2000&=200(1.07)^t\\10&=(1.07)^t\\\\\textsf{Take natural logs:} \quad \ln (10)&=\ln (1.07)^t\\\ln (10)&=t\ln (1.07)\\t&=\dfrac{\ln (10)}{\ln (1.07)}\\t&=34.0323838...\end{aligned}[/tex]
The value of the baseball card will reach $2,000 during the 34th year - after 34 years and 12 days. As this is not an investment account where interest is applied annually, we don't need to round up. Therefore, the baseball card will be worth $2,000 after 34 years and 12 days.
