Problem 12 - Applied Calculus

Question

If \(t\) is measured in days since June 1 , the inventory \(I(t)\) for an item in a warehouse is given by $$ I(t)=5000(0.9)^{t} $$ (a) Find the average inventory in the warehouse during the 90 days after June 1 . (b) Graph \(I(t)\) and illustrate the average graphically.

Step-by-Step Solution

Answer

The average inventory over 90 days is approximately 2635 items.

Step 1: Understand the Formula

The inventory function given is \(I(t) = 5000(0.9)^t\). This represents the number of items in a warehouse, where \(t\) is the number of days since June 1.

Step 2: Calculate the Integral of I(t)

To find the average inventory over 90 days, we need to integrate the function \(I(t)\) from \(t=0\) to \(t=90\). The integral of \(I(t)\) is the function:\[ \int_0^{90} 5000(0.9)^t \, dt \]This requires the use of substitution or a table of integrals for exponential functions.

Step 3: Apply the Exponential Integral

The integral of \(a^t\) is given by \(\frac{a^t}{\ln(a)}\). Applying this to our function, we use \(a=0.9\) to obtain:\[ \int 5000(0.9)^t \, dt = \frac{5000(0.9)^t}{\ln(0.9)} \] evaluated from 0 to 90.

Step 4: Evaluate the Definite Integral

Now we evaluate:\[ \left[ \frac{5000(0.9)^t}{\ln(0.9)} \right]_0^{90} = \frac{5000(0.9)^{90}}{\ln(0.9)} - \frac{5000(0.9)^{0}}{\ln(0.9)} \]Calculate each part to find the definite integral value.

Step 5: Calculate the Average Inventory

The average value of the function over the interval \([0, 90]\) is given by:\[ \text{Average} = \frac{1}{90} \cdot \left(\frac{5000(0.9)^{90}}{\ln(0.9)} - \frac{5000(0.9)^{0}}{\ln(0.9)}\right) \]Calculate the numeric value for the average inventory.

Step 6: Sketch the Graph of I(t)

Plot the graph of \(I(t) = 5000(0.9)^t\) using a graphing tool or by hand. The graph will show an exponentially decaying function starting at 5000.

Step 7: Illustrate the Average on the Graph

On the graph, draw a horizontal line at the average inventory value computed in Step 5. This line represents the average inventory level over 90 days.

Key Concepts

Average Value Theorem

The Average Value Theorem helps us find the average value of a function over a specific interval. In simple terms, it's like finding the "mean" or "central" value of a function. For a function \( f(x) \), the average value on the interval \[ [a, b] \] is given by:\[\text{Average} = \frac{1}{b-a} \int_a^b f(x) \ dx.\]In the context of our exercise, we're finding the average inventory, \( I(t) \), in a warehouse over the 90-day period starting from June 1. By computing the definite integral of \( I(t) \) and dividing by 90, we determine how many items, on average, are held in inventory each day. This method provides a single value summarizing inventory levels over time, smoothing out day-to-day fluctuations.

Definite Integral

A definite integral computes the "net area" under a curve, offering a precise accumulation of quantities such as inventory, cost, or distance. In our example, we compute:\[\int_{0}^{90} 5000(0.9)^t \ dt.\]This integral represents the area under the curve of \( I(t) = 5000(0.9)^t \) from day 0 to day 90. To calculate this, you often rely on known rules for integrating specific types of functions, such as exponential functions. The result of this integral gives you total inventory over these days before you average it with the Average Value Theorem. This approach thoroughly considers daily changes in the inventory, unlike a simple arithmetic mean which wouldn't recognize such variations.

Exponential Functions

Exponential functions, like \( I(t) = 5000(0.9)^t \), are powerful tools to model processes where quantities decrease or increase at a constant percentage rate. Here, \( (0.9)^t \) indicates exponential decay, meaning the variable quantity decreases by 10% each day.Key features of exponential functions include:

• A constant base that raises a variable exponent.
• Rapid initial change, followed by slower rates as you progress along the function.
• Real-world applications in finance, biology, physics, and inventory management.

Understanding these functions allows us to accurately predict and examine how processes like inventory decline in a warehouse due to sales over time.

Graphing Functions

Graphing is an essential skill that translates mathematical functions into visual understanding. For \( I(t) = 5000(0.9)^t \), graphing reveals the nature of exponential decay visually.Steps to graph \( I(t) \):

• Identify initial value, which is \( 5000 \), when \( t = 0 \).
• Calculate further points by plugging values of \( t \) (e.g., \( t = 10, 20, 30... \)) into the equation.
• Plot these points on the Cartesian plane.
• Draw a smooth curve through the points, showcasing a continually decreasing quantity.

To visualize the average effectively, add a horizontal line at the average inventory level across the graph. This line helps to compare and understand the fluctuations in real time versus the average condition."