Exponential growth word problem 

Since the growth is exponential, 
$$V(t)=V_0 e^{a t}=269 e^{kt}.$$

a)
$$V(9)=269e^{9 k}=477      \Rightarrow    e^{9k}=\frac{477}{269}    \Rightarrow 9k=\ln \frac{477}{269} .$$
So
$$k=\frac{1}{9}\ln (\frac{477}{269})=0.064.$$ 

b) $V(t)=269 e^{0.064t}.$
c) We have 2013-1980=33. 

$$v(33)=269 e^{0.064 \times 33}=269*8.17=\$ 2196.7.$$

d) To find the doubling time we must have $V(t)=2v(0)$. Hence
$$269 e^{kt}=2 \times 269  \Rightarrow e^{kt}=2  \Rightarrow kt= \ln 2 $$
\[\Rightarrow t=\frac{\ln 2}{k}=\frac{\ln 2}{0.064}=10.8.\]

e) \[V(t)=269 e^{kt}=2878  \Rightarrow e^{kt}=\frac{2878}{269}  \Rightarrow kt =\ln (\frac{2878}{269})=2.37.\]
Hence
\[0.064 t= 2.37  \Rightarrow t= 37.03. \]
So the value of the coin will be $\$ 2878$ in the year 1980+37.03=2017.03.