Problem 12 - Calculus and its applications

Question

Annual net sales. Green Mountain Coffee Roasters produces many varieties of flavored coffees, teas, and K-cups. Since \(2008,\) the net sales \(S\) of the company have grown exponentially at the rate of \(46.3 \%\) per year. This growth can be approximated by $$ \frac{d S}{d t}=0.463 S $$ where \(t\) is the number of years since 2008 . (Source: Green Mountain Coffee Roasters financial statements.) a) Find the function that satisfies the equation, given that net sales in \(2008(t=0)\) were approximately \(\$ 500.000\) b) Estimate net sales in 2012 and 2016 . c) What is the doubling time for \(S(t) ?\)

Step-by-Step Solution

Answer

a) \( S(t) = 500,000e^{0.463t} \); b) 2012: \( \$2,310,282 \), 2016: \( \$10,676,280 \); c) Doubling time: \( 1.496 \) years.

Step 1: Solve the Differential Equation

The given differential equation is \( \frac{dS}{dt} = 0.463S \). This is a separable equation, and it can be solved by separating variables as follows: \( \frac{dS}{S} = 0.463 \, dt \). Integrating both sides gives \( \ln |S| = 0.463t + C \), where \( C \) is the constant of integration.

Step 2: Exponentiate Both Sides

Exponentiating both sides of the equation from Step 1 yields \( S = e^{0.463t + C} = e^C e^{0.463t} \). Letting \( A = e^C \), we have \( S(t) = A e^{0.463t} \). This gives the general solution of the differential equation.

Step 3: Determine the Constant Using Initial Condition

We know that in 2008 (\( t = 0 \)), net sales were \( \$500,000 \). Using this information, substitute \( t = 0 \) and \( S = 500,000 \) into the equation: \( 500,000 = Ae^{0.463 \times 0} = A \). Therefore, \( A = 500,000 \). The specific solution is \( S(t) = 500,000e^{0.463t} \).

Step 4: Estimate Net Sales for 2012

To estimate net sales in 2012, calculate \( t = 2012 - 2008 = 4 \). Substitute \( t = 4 \) into the specific solution: \( S(4) = 500,000e^{0.463 \times 4} \). Calculating gives \( S(4) \approx 500,000e^{1.852} \approx 2,310,282 \).

Step 5: Estimate Net Sales for 2016

To estimate net sales in 2016, calculate \( t = 2016 - 2008 = 8 \). Substitute \( t = 8 \) into the specific solution: \( S(8) = 500,000e^{0.463 \times 8} \). Calculating gives \( S(8) \approx 500,000e^{3.704} \approx 10,676,280 \).

Step 6: Calculate Doubling Time

The doubling time \( T_d \) can be found from the equation \( 2 = e^{0.463T_d} \). Taking the natural logarithm of both sides gives \( \ln 2 = 0.463T_d \), resulting in \( T_d = \frac{\ln 2}{0.463} \). Calculating gives \( T_d \approx 1.496 \) years.

Key Concepts

Differential Equations

A differential equation involves functions and their derivatives, showing how these functions change. In our exercise, the equation \( \frac{dS}{dt} = 0.463S \) describes the exponential growth of net sales over time. Here, \( S \) represents the sales, and \( \frac{dS}{dt} \) is the rate at which sales change concerning time \( t \).

This equation is what's known as a **separable differential equation**. It means we can rearrange it to a form that allows each variable and its differential to be on separate sides. By doing so, we separate \( S \) and \( t \) and solve the equation through integration, leading us to integrate \( \frac{dS}{S} = 0.463 \, dt \). The integration provides a natural logarithm on one side and a linear expression in \( t \) on the other.

The integration results in the equation \( \ln |S| = 0.463t + C \), which requires exponentiating to solve for \( S \). This yields \( S = e^{0.463t + C} \) or, simplified, \( S = Ae^{0.463t} \), where \( A = e^C \). This is the general solution indicating the exponential relationship between sales and time.

Net Sales Estimation

Estimating future net sales involves substituting values into our derived function \( S(t) = 500,000e^{0.463t} \). This equation helps predict the company's future sales based on past growth patterns.

### Estimating Sales in 2012 and 2016To forecast net sales:

• For 2012, calculate \( t = 4 \), since 2012 is 4 years after 2008. Plugging this into the equation gives \( S(4) = 500,000e^{0.463 \times 4} \), which is approximately \( 2,310,282 \).
• For 2016, calculate \( t = 8 \), since 2016 is 8 years after 2008. Plug this into the equation to find \( S(8) = 500,000e^{0.463 \times 8} \), estimating \( 10,676,280 \).

These calculations show us how exponential growth affects large gains in sales numbers over short periods. It demonstrates the potency of compounding growth rates in financial estimations.

Doubling Time Calculation

Doubling time refers to the period it takes for an amount to double in size or value. For sales, it is a direct indicator of growth speed.

To find the doubling time for our exponential growth equation, solve \( 2 = e^{0.463T_d} \), where \( T_d \) is the doubling time. Taking the natural logarithm of both sides gives \( \ln 2 = 0.463T_d \). Solving for \( T_d \) results in:

• \( T_d = \frac{\ln 2}{0.463} \)
• Approximation yields \( T_d \approx 1.496 \) years.

This result shows that with the given exponential growth rate, Green Mountain Coffee Roasters' net sales double approximately every 1.5 years. Understanding doubling time highlights the significant impact of continuous growth, emphasizing the rapid increase in future sales projections.