A particular country's exports of goods are increasing exponentially. The value of the exports, t...

Question:

A particular country's exports of goods are increasing exponentially. The value of the exports, t years after 2009, can be approximated by {eq}V(t)=1.7e^{0.049t} {/eq} where t = 0 corresponds to 2009, and V is in billions of dollars.

a) Estimate the value of the country's exports in 2009 and 2014.

b) What is the doubling time for the value of the country's exports?

c) What is the value of the country's exports in 2009?

d) What is the value of the country's exports in 2014?

e) The doubling time is approximately how many years?

Exponential Functions:

Represented mathematically as {eq}P(t) = P(0)e^{rt} {/eq}, exponential growth models can be applied to describe the population increase over time. In this model, {eq}P(t) {/eq} indicates the population at time {eq}t {/eq}, {eq}P(0) {/eq} represents the initial population, {eq}r {/eq} reflects the growth rate, and {eq}t {/eq} is the specific point in time.

Given: {eq}V(t) = 1.7e^{0.049t} \\ \text{ 2009 }: t = 0 {/eq}

a. To estimate the value of the country's exports in {eq}2009 {/eq} and {eq}2014 {/eq} (which occurs {eq}5 {/eq} years after {eq}2009 {/eq}), the strategy is to evaluate the function {eq}V(t) {/eq} when {eq}t = 0 {/eq} and {eq}t = 5 {/eq} respectively.

In {eq}2009 {/eq}:

{eq}\begin{align*} V(0) &= 1.7e^{0.049(0)} \\ &= 1.7e^{0} \\ &= 1.7(1) \\ &= 1.7 \\ \end{align*} {/eq}

Therefore, the value of the country's exports in {eq}2009 {/eq} is approximately {eq}1.7 {/eq} billion dollars.

In {eq}2014 {/eq}:

{eq}\begin{align*} V(5) &= 1.7e^{0.049(5)} \\ &= 1.7e^{0.245} \\ &= 1.7(1.277621313) \\ &= 2.171956232 \\ \end{align*} {/eq}

Therefore, the value of the country's exports in {eq}2014 {/eq} is approximately {eq}2.172 {/eq} billion dollars.

b. The doubling time for the value of the country's exports occurs at the time when the value of the country's exports is doubled, or two times the initial value of the country's exports. Based on the exponential model ({eq}V(t) = V(0)e^{rt} {/eq}), this can be seen when {eq}V(t) = 2V(0) {/eq}. In this case, the calculation {eq}V(t) = 2(1.7) = 3.4 {/eq} best represents this. Therefore, the strategy is to substitute the condition {eq}V(t) = 3.4 {/eq} in the exponential equation {eq}V(t) = 1.7e^{0.049t} {/eq} and solve for the unknown {eq}t {/eq} variable.

{eq}\begin{align*} 3.4 = 1.7e^{0.049t} &\Rightarrow 3.4\div 1.7 = 1.7e^{0.049t}\div 1.7 \\ &\Rightarrow 2 = e^{0.049t} \\ &\Rightarrow \ln(2) = \ln(e^{0.049t}) \\ &\Rightarrow 0.6931471806 = 0.049t \\ &\Rightarrow 0.6931471806\div 0.049 = 0.049t\div 0.049 \\ &\Rightarrow t = 14.14586083 \\ \end{align*} {/eq}

Therefore, the doubling time for the value of the country's exports is approximately {eq}14.146 {/eq} years.