A particular country's exports of goods are increasing exponentially. The value of the exports, t years after 2005, can be approximated by [tex]V ( t )=1.4 e^{0.043 t }[/tex] where [tex]t =0[/tex] corresponds to 2005 and V is in billions of dollars.
a) Estimate the value of the country's exports in 2005 and 2015.
b) What is the doubling time for the value of the country's exports?
a) The value of the country's exports in 2005 is $1.4 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 2015 is $ [ ] billion.
[ ]
(Simplify your answer. Round to the nearest tenth as needed. Do not include the $ symbol in your answer.)
Answer (1)
Calculate the value of exports in 2015 by substituting t = 10 into the formula: V ( 10 ) = 1.4 e 0.043 × 10 a pp ro x 2.2 .
Determine the doubling time by setting V ( t ) = 2.8 : 2.8 = 1.4 e 0.043 t .
Solve for t by taking the natural logarithm: ln ( 2 ) = 0.043 t , which gives t = 0.043 l n ( 2 ) a pp ro x 16.1 .
The value of exports in 2015 is approximately 2.2 billion dollars and the doubling time is approximately 16.1 years: 16.1 .
Explanation
Understanding the Problem We are given the formula for the value of a country's exports as a function of time: V ( t ) = 1.4 e 0.043 t , where t is the number of years after 2005 and V ( t ) is in billions of dollars. We need to find the value of the exports in 2015 and the doubling time for the exports.
Calculating Exports in 2015 First, let's find the value of the exports in 2015. Since t is the number of years after 2005, for the year 2015, t = 2015 − 2005 = 10 . We substitute t = 10 into the formula: V ( 10 ) = 1.4 e 0.043 × 10 .
Finding the Value Now, we calculate the value: V ( 10 ) = 1.4 e 0.43 . Using a calculator, we find that e 0.43 a pp ro x 1.53733 . Therefore, V ( 10 ) a pp ro x 1.4 × 1.53733 a pp ro x 2.152 . Rounding to the nearest tenth, we get V ( 10 ) a pp ro x 2.2 billion dollars.
Understanding Doubling Time Next, we need to find the doubling time. The doubling time is the time it takes for the value of the exports to double. Since the initial value in 2005 ( t = 0 ) is 1.4 billion dollars, we want to find the time t when V ( t ) = 2 × 1.4 = 2.8 billion dollars.
Setting up the Equation We set up the equation 2.8 = 1.4 e 0.043 t . Dividing both sides by 1.4 , we get 2 = e 0.043 t . To solve for t , we take the natural logarithm of both sides: ln ( 2 ) = ln ( e 0.043 t ) , which simplifies to ln ( 2 ) = 0.043 t .
Solving for Time Now, we solve for t : t = 0.043 l n ( 2 ) . Using a calculator, we find that ln ( 2 ) a pp ro x 0.69315 . Therefore, t a pp ro x 0.043 0.69315 a pp ro x 16.12 . Rounding to the nearest tenth, we get t a pp ro x 16.1 years.
Final Answer Therefore, the value of the country's exports in 2015 is approximately 2.2 billion dollars, and the doubling time for the value of the country's exports is approximately 16.1 years.
Examples
Exponential growth models are used in various real-world scenarios, such as predicting population growth, calculating compound interest, and modeling the spread of diseases. For instance, understanding the doubling time of an investment helps investors estimate how long it will take for their money to double at a given interest rate. Similarly, in epidemiology, knowing the doubling time of an infectious disease helps public health officials implement timely interventions to control its spread. These models provide valuable insights for making informed decisions in finance, healthcare, and other fields.
