Problem 41 - Applied Calculus

Question

The Hershey Company is the largest US producer of chocolate. In \(2011,\) annual net sales were 6.1 billion dollars and were increasing at a continuous rate of \(4.2 \%\) per year. \(^{65}\) (a) Write a formula for annual net sales, \(S,\) as a function of time, \(t,\) in years since 2011 (b) Estimate annual net sales in 2015 (c) Use a graph to estimate the year in which annual net sales are expected to pass 8 billion dollars and check your estimate using logarithms.

Step-by-Step Solution

Answer

(a) \( S(t) = 6.1 \cdot e^{0.042t} \); (b) \( S(4) \approx 7.233 \text{ billion} \); (c) Sales surpass 8 billion around 2019.

Step 1: Determine the exponential growth formula

To model annual net sales as a function of time, let's start by using the general formula for continuous exponential growth: \( S(t) = S_0 \cdot e^{rt} \). Here, \( S_0 \) is the initial amount (6.1 billion dollars in 2011), \( r \) is the growth rate (4.2%, or 0.042 when expressed as a decimal), and \( t \) is the time in years since 2011. The formula becomes \( S(t) = 6.1 \cdot e^{0.042t} \).

Step 2: Calculate annual net sales in 2015

To find the net sales in 2015, plug \( t = 4 \) (since 2015 is 4 years after 2011) into the formula. Calculate \( S(4) = 6.1 \cdot e^{0.042 \times 4} \). Evaluating the exponent and then the formula gives us the sales for 2015.

Step 3: Analyze graphically the year net sales surpass 8 billion

To estimate graphically the year when sales pass 8 billion dollars, plot the function \( S(t) = 6.1 \cdot e^{0.042t} \) and find the point where this function exceeds 8. On the graph, identify the value of \( t \) where \( S(t) > 8 \).

Step 4: Verify using logarithms when net sales will pass 8 billion

To solve algebraically, set \( S(t) = 8 \) and solve for \( t \): \( 8 = 6.1 \cdot e^{0.042t} \). First, divide both sides by 6.1 to isolate the exponential term: \[ e^{0.042t} = \frac{8}{6.1} \]. Then take the natural logarithm of both sides: \[ \ln(e^{0.042t}) = \ln\left(\frac{8}{6.1}\right) \]. This simplifies to: \( 0.042t = \ln\left(\frac{8}{6.1}\right) \). Solve for \( t \) by dividing by 0.042: \( t = \frac{\ln\left(\frac{8}{6.1}\right)}{0.042} \). This gives the year when sales surpass 8 billion dollars.

Key Concepts

Calculus Problem Solving

Calculus helps us solve many real-world problems, especially those involving rates and continuous changes, like exponential growth. In this exercise, we're modeling Hershey's annual net sales over time. This involves understanding how quantities change continuously. Here, we use the continuous exponential growth formula:

• Understand the general formula: \( S(t) = S_0 \cdot e^{rt} \), where \( S_0 \) is the initial amount, \( r \) is the continuous growth rate, and \( t \) represents time.
• Substitute given values: With \( S_0 = 6.1 \) billion and \( r = 0.042 \) (as a decimal), the formula becomes \( S(t) = 6.1 \cdot e^{0.042t} \).
• Solving for specific years: To estimate sales in 2015, set \( t = 4 \) since 2015 is 4 years after 2011, and solve \( S(4) = 6.1 \cdot e^{0.042 \times 4} \).

This way of solving uses calculus principles to manage and predict growth rates effectively, showing the power of mathematical modeling in business contexts. Always ensure you have the correct initial conditions and growth rates for precise calculations.

Logarithmic Functions

Logarithms, a key concept in mathematics, are incredibly useful for solving equations involving exponential growth, especially when reversing or estimating growth. In this exercise, after finding a time point where net sales reach a particular target, logarithms become essential.

• Understand the process: To estimate when sales surpass a certain figure, set the formula equal to that value, for instance, \( 8 = 6.1 \cdot e^{0.042t} \).
• Simplify using logarithms: Divide by the initial sales, then use natural logarithms to cancel out the exponent. This becomes \( e^{0.042t} = \frac{8}{6.1} \) and then \( 0.042t = \ln\left(\frac{8}{6.1}\right) \).
• Compute the solution: Solve for \( t \) to find the exact year when sales exceed 8 billion, using \( t = \frac{\ln\left(\frac{8}{6.1}\right)}{0.042} \).

Logarithmic functions thus provide an analytical method to determine growth timelines without having to graph functions visually every time. They simplify and enhance our solution processes.

Financial Modeling

Financial modeling involves creating representations of a financial scenario, which includes predicting future data points like revenues. Here, with Hershey's net sales growth, financial modeling integrates exponential functions and provides insights into future possibilities.

• Create a realistic growth model: Use historical data and reasonable growth assumptions. Here, we begin with \( S_0 = 6.1 \) billion and a growth rate of \( 4.2\% \).
• Forecast future values: Using the exponential growth formula, predict future sales figures at any given time. This can inform business strategies and decision-making.
• Evaluate business milestones: Determine when goals, like surpassing 8 billion dollars in sales, will be met. Use both graphical and logarithmic analysis to confirm predictions.

Such models are invaluable for businesses in planning and strategic development. They ensure firms are well-prepared for future expansions or adjustments.