Questions: A particular country's exports of goods are increasing exponentially. The value of the exports, t years after 2006, can be approximated by V(t)=1.3 e^0.0071 where t=0 corresponds to 2006 and V is in billions of dollars. a) Estimate the value of the country's exports in 2006 and 2012 b) What is the doubling time for the value of the country's exports? a) The value of the country's exports in 2006 is billion. (Simplify your answer. Round to the nearest tenth as needed. Do not include the symbol in your answer.) The value of the country's exports in 2012 is 5 billion. (Simplify your answer. Round to the nearest tenth as needed. Do not include the symbol in your answer.) b) The doubling time is approximately years. (Simplify your answer. Round to the nearest integer as needed.)

The estimated value of the exports after 0 years is approximately 1.3 billion dollars, and the doubling time for the value of these exports is approximately 97.6 years.

To estimate the value of the country's exports $t$ years after the base year, we use the formula $V(t) = P e^{rt}$, where $P$ is the initial value of the exports in the base year, $r$ is the rate of growth per year, and $t$ is the number of years after the base year. Substituting the given values into the formula, we get $V(t) = 1.3 e^{0.0071(6)} = 1.4$ billion dollars.

The doubling time is found by setting $V(t) = 2P$ and solving for $t$. This gives $2 = e^{rt}$. Taking the natural logarithm of both sides gives $\ln(2) = rt$, and solving for $t$ gives $t = \frac{\ln(2)}{r}$. Substituting the given rate of growth $r = 0.0071$ into the formula, we find the doubling time to be $t = \frac{\ln(2)}{0.0071} = 97.6$ years.

The estimated value of the exports after 6 years is approximately 1.4 billion dollars, and the doubling time for the value of these exports is approximately 97.6 years.

To estimate the value of the country's exports $t$ years after the base year, we use the formula $V(t) = P e^{rt}$, where $P$ is the initial value of the exports in the base year, $r$ is the rate of growth per year, and $t$ is the number of years after the base year. Substituting the given values into the formula, we get $V(t) = 1.3 e^{0.0071(0)} = 1$ billion dollars.

The doubling time is found by setting $V(t) = 2P$ and solving for $t$. This gives $2 = e^{rt}$. Taking the natural logarithm of both sides gives $\ln(2) = rt$, and solving for $t$ gives $t = \frac{\ln(2)}{r}$. Substituting the given rate of growth $r = 0.0071$ into the formula, we find the doubling time to be $t = \frac{\ln(2)}{0.0071} = 98$ years.

The estimated value of the exports after 0 years is approximately 1 billion dollars, and the doubling time for the value of these exports is approximately 98 years.