Problem 81 - Calculus and its applications

Question

U.S. travel exports (goods and services that international travelers buy while visiting the United States) are increasing exponentially. The value of such exports, \(t\) years after \(2011,\) can be approximated by $$ V(t)=115.32 e^{0.094 t} $$ where \(V\) is in billions of dollars. (Source: www.census. gov/foreign- trade/data/index.html.) a) Estimate the value of U.S. travel exports in 2016 and 2018 b) Estimate the growth rate for U.S. travel exports in 2016 and 2018

Step-by-Step Solution

Answer

In 2016, exports were about $184.47 billion; in 2018, $222.52 billion. The growth rate is 9.4% per year.

Step 1: Identify the known values

To find the value of U.S. travel exports for a specific year, identify the value of \( t \) for each year. For 2016, \( t = 2016 - 2011 = 5 \). For 2018, \( t = 2018 - 2011 = 7 \).

Step 2: Substitute into the exponential formula for 2016

Using the formula \( V(t) = 115.32 e^{0.094 t} \), substitute \( t = 5 \) to estimate exports in 2016: \[ V(5) = 115.32 e^{0.094 \times 5} \].

Step 3: Calculate the exponential expression for 2016

Calculate: \[ e^{0.094 \times 5} = e^{0.47} \approx 1.5992 \].

Step 4: Compute the export value for 2016

Multiply the base value by the exponential result: \[ V(5) = 115.32 \times 1.5992 \approx 184.472 \]. Thus, the estimated value for 2016 is approximately $184.47 billion.

Step 5: Substitute into the exponential formula for 2018

Using the same formula, substitute \( t = 7 \) to estimate exports in 2018: \[ V(7) = 115.32 e^{0.094 \times 7} \].

Step 6: Calculate the exponential expression for 2018

Calculate: \[ e^{0.094 \times 7} = e^{0.658} \approx 1.9310 \].

Step 7: Compute the export value for 2018

Multiply the base value by the exponential result: \[ V(7) = 115.32 \times 1.9310 \approx 222.523 \]. Thus, the estimated value for 2018 is approximately $222.52 billion.

Step 8: Calculate the growth rate for 2016

The growth rate can be found by the exponent coefficient: the growth rate is approximately \( 0.094 \) or 9.4% per year, constant from the equation.

Step 9: Confirm the growth rate for 2018

Since the exponential growth model uses a constant rate, the annual growth rate for 2018 remains the same at approximately 9.4%.

Key Concepts

U.S. Travel Exports

U.S. travel exports refer to the goods and services purchased by international travelers who visit the United States. These exports are a significant economic contributor, as tourists spend money on experiences, accommodation, food, transportation, and other purchases. Understanding their growth is essential for economic forecasting and policy making.
U.S. travel exports are typically measured in billions of dollars, indicating the size and impact of the travel industry on the nation's economy. By exporting travel and tourism services, the U.S. generates income from external sources, boosting its trade balance.
The increasing numbers in travel exports can be attributed to several factors:

• An attractive array of tourist destinations
• Advanced infrastructure
• Friendly policies towards international visitors

This industry is sensitive to global economic conditions, exchange rates, and international relations, which can influence the level of foreign tourists and their spending habits.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It is typically used to model growth processes, such as populations, investments, or, in this case, U.S. travel exports.
The formula provided in the exercise is:\[ V(t) = 115.32 \, e^{0.094 t} \]
This formula indicates how the value of U.S. travel exports (\(V(t)\)) changes over time (\(t\)). The base of the natural exponential function is \(e\), approximately equal to 2.71828, which is used to model continuous growth scenarios.
In this formula:

• The number 115.32 represents the initial value of exports in billions of dollars in 2011.
• The exponent\(0.094t\)consists of a rate of growth \(0.094\)and the number of years \(t\), demonstrating how exports grow each year.

Exponential growth is characterized by a constant percentage increase per time period. In this case, U.S. travel exports grow by approximately 9.4% each year.

Growth Rate Calculation

Growth rate calculation involves determining how quickly a quantity is increasing over time. In exponential functions, this growth is often expressed as a percentage that remains constant every year.
In the context of the formula for U.S. travel exports, the growth rate can be directly identified from the exponent of the function, which is 9.4%.
When calculating growth rates:

• Understanding that the growth rate is constant helps in predicting future values without recalculating each year.
• The formula itself, \(V(t) = 115.32\, e^{0.094 t}\), simplifies the calculation of future export values, as each year's value builds on the previous ones.

For example, by setting different values of \(t\), one can compute the exports for different years and understand the compound nature of the exponential increase.
Predicting future scenarios depends on assuming continued consistency in external variables, like global economic conditions, which can, however, alter actual growth trends.