Problem 11 - Applied Calculus

Question

If \(t\) is measured in days since June 1 , the inventory \(I(t)\) for an item in a warchouse 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 a result derived from the integral, approximately computed numerically. Graphically, the average line can be visually represented over the decay curve of the inventory function.

Step 1: Determine the Function for Average Inventory

To find the average inventory over a specific period, you would integrate the inventory function over the given time and divide by the length of the time period. The formula for the average value of a function over an interval \([a, b]\) is given by: \[ \text{Avg}(I(t)) = \frac{1}{b-a} \int_{a}^{b} I(t) \, dt \] In this case, we need to find the average inventory from \(t=0\) to \(t=90\).

Step 2: Set Up and Evaluate the Integral

First, set up the integral of the given function over the interval from 0 to 90: \[ \int_{0}^{90} 5000(0.9)^t \, dt \] To evaluate this, we take out the constant 5000 and apply the integral formula for exponential functions: \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \). This becomes: \[ 5000 \int_{0}^{90} (0.9)^t \, dt = 5000 \Bigg[ \frac{(0.9)^t}{\ln(0.9)} \Bigg]_{0}^{90} \]

Step 3: Calculate the Definite Integral

Calculate the definite integral by substituting the limits into the expression: \[ 5000 \left( \frac{(0.9)^{90}}{\ln(0.9)} - \frac{(0.9)^{0}}{\ln(0.9)} \right) \] Using a calculator to find \((0.9)^{90}\) and \(\ln(0.9)\), you can numerically solve this to find the integral value.

Step 4: Find the Average Inventory

Divide the result of the integral by the time period to find the average inventory: \[ \text{Avg}(I(t)) = \frac{5000 \left( \frac{(0.9)^{90}}{\ln(0.9)} - \frac{(0.9)^{0}}{\ln(0.9)} \right)}{90} \] Solve numerically to find the average.

Step 5: Graph the Function and Show the Average

The function \(I(t) = 5000(0.9)^t\) is an exponential decay function. To graph it, plot \(t\) on the x-axis and \(I(t)\) on the y-axis from \(t=0\) to \(t=90\). Draw a horizontal line at the average inventory value found in the previous step to represent the average inventory over the period.

Key Concepts

Average Value of a Function

The average value of a function gives us an insightful overview of a function's behavior across a specific interval. Imagine you have a roller coaster. Calculating the average height can tell you a lot about the typical experience you might have during a ride.

For an inventory problem like ours, finding the average inventory of items over a period is vital for planning and logistics. The average value of a function over an interval \([a, b]\) is computed using this formula:

• \( \text{Avg}(I(t)) = \frac{1}{b-a} \int_{a}^{b} I(t) \, dt \)

Here:

• \( I(t) \) is the inventory function
• \( a \) and \( b \) are the start and end of the interval, respectively

For the given problem, your \( a \) is 0 and \( b \) is 90 days. This calculation collates the daily inventory over 90 days, giving you the mean inventory level. Its practical significance is immense: it helps in understanding how stocked the warehouse stays on average over summer.

Definite Integral

A definite integral is a fundamental concept in calculus, mostly used to find the area under a curve. It is written as \( \int_{a}^{b} f(x) \, dx \). The integral gives a precise accumulation of quantities, much like how you accumulate scores in a video game.

In the context of inventory, our aim with a definite integral is to determine the total inventory over a specified time period. Here, we'll use the function \( I(t) = 5000(0.9)^t \), which represents a typical exponential decay as items might deplete over time due to factors like sales or spoilage.

In our specific example, setting up the integral correctly is crucial:

• \( \int_{0}^{90} 5000(0.9)^t \, dt \)

By evaluating this integral, we compute the comfort of having a definite measure of total items that cycle through until restocking or another event happens. After integration and applying limits, you get a precise numerical output that can be used to find an average or track resource depletion over time.

Graphing Functions

Graphing functions is a critical skill for visualizing mathematical relationships. It allows us to see the changes in quantities and their rate over time, making abstract concepts tangible.

For our inventory function, \( I(t) = 5000(0.9)^t \), graphing provides two functionalities:

• Showing how inventory decreases over time due to natural losses or reductions
• Clearly illustrating the average inventory level across the specified period

To graph this, plot \( t \) (time in days) on the x-axis and \( I(t) \) (inventory) on the y-axis. You'll notice a downward trend, typical of exponential decay functions, where the rate of decrease slows over time.

Put a horizontal line at the calculated average inventory value. This line acts as a visual reference across the graphing period, helping quickly show how often and by how much the actual inventory dips below or above this average. Visualizing it this way can aid in making quick and effective decisions about when to reorder or assess the storage capacity and needs.