Problem 19 - Applied Calculus

Question

The rate of sales (in sales per month) of a company is given, for \(t\) in months since January 1 , by $$ r(t)=t^{4}-20 t^{3}+118 t^{2}-180 t+200 . $$ (a) Graph the rate of sales per month during the first year \((t=0\) to \(t=12\) ). Does it appear that more sales were made during the first half of the year, or during the second half? (b) Estimate the total sales during the first 6 months of the year and during the last 6 months of the year. (c) What are the total sales for the entire year? (d) Find the average sales per month during the year.

Step-by-Step Solution

Answer

(a) Visually analyze the graph to determine the comparison. (b) Integrate to estimate. (c) Integrate over 12 months. (d) Divide total sales by 12.

Step 1: Understand the Function

The function given, \( r(t) = t^{4} - 20t^{3} + 118t^{2} - 180t + 200 \), represents the rate of sales in terms of months since January 1st. Our tasks are to analyze this function over a 12-month period.

Step 2: Plot the Rate of Sales

Graph the function \( r(t) \) over the interval \( t = 0 \) to \( t = 12 \). Observe the trend of the graph to compare sales in the first half of the year (\( t=0 \) to \( t=6 \)) with the second half (\( t=6 \) to \( t=12 \)). This can be done using graphing tools such as Desmos or a graphing calculator.

Step 3: Observe the Graph

After plotting, the graph will show distinct behavior in the sales rate. We can visually compare the areas under the curve for the first and second half of the year to determine which half has greater sales volume. Typically, using calculus would be better, but a visual examination provides a clear impression.

Step 4: Set Up Integrals for Total Sales Estimation

To estimate the total sales for each half of the year, integrate the function \( r(t) \) over the intervals \( [0, 6] \) and \( [6, 12] \). These integrals are the exact calculation of the total sales for each period.\[\int_{0}^{6} r(t) \, dt \quad \text{and} \quad \int_{6}^{12} r(t) \, dt\]

Step 5: Calculate the First Half-Year Sales

Calculate \( \int_{0}^{6} (t^{4} - 20t^{3} + 118t^{2} - 180t + 200) \, dt \) to find the total sales in the first half of the year. This can be done using definite integration techniques or numerical methods if a calculator is used.

Step 6: Calculate the Second Half-Year Sales

Find \( \int_{6}^{12} (t^{4} - 20t^{3} + 118t^{2} - 180t + 200) \, dt \) to calculate total sales in the second half of the year. Similarly, use definite integration methods.

Step 7: Total Sales for the Year

Add the results from Steps 5 and 6 to find the total sales for the year:\[\text{Total Sales} = \int_{0}^{12} r(t) \, dt\]Carry out the calculation to compute the total.

Step 8: Average Sales per Month

Divide the total sales from Step 7 by 12 to find the average sales per month:\[\text{Average Sales per Month} = \frac{1}{12} \int_{0}^{12} r(t) \, dt\]

Key Concepts

Rate of Sales

The rate of sales function, denoted as \( r(t) \), is crucial because it tells us how quickly products are being sold over time. In this case, \( r(t) = t^4 - 20t^3 + 118t^2 - 180t + 200 \) describes how sales change month by month, starting from January. Here, \( t \) represents the number of months since January, so \( t=0 \) would be January, \( t=1 \) would be February, and so on.

Understanding the rate of sales gives us insight into business performance. Higher values of \( r(t) \) indicate more brisk sales, while lower values suggest a slump. By analyzing the function's peaks, troughs, and overall trend, businesses can determine if they are hitting their monthly targets or if adjustments in strategy are needed.

Plotting the function can help visualize when sales were most active. For this function specifically, students are asked to observe which half of the year had better sales performance: the first six months or the last. By merely looking at the graph of \( r(t) \), one can quickly get a visual summary of trends and changes in sales dynamics.

Integration in Sales Analysis

Integration is a powerful tool used to calculate total quantities from rates. In this context, to find the total sales in a given period, we integrate the rate of sales function \( r(t) \) over the intervals of interest. This process gives us the area under the curve of \( r(t) \), which corresponds to the total sales.

Here is how integration helps:

• To find total sales for the first half of the year, compute the definite integral \( \int_{0}^{6} r(t) \, dt \).
• For the second half, the integral is \( \int_{6}^{12} r(t) \, dt \).

Using these integrals, students can calculate the precise total sales rather than relying on visual approximations from graph plots. Calculating these ensures that each month's contribution to the annual sales is accurately included.

In practical terms, proper integration allows businesses to measure performance over time accurately. Companies can adjust forecasts, manage inventories, or even revise strategies based on the actual sales computed from these integrals.

Average Sales

To understand average sales, one must calculate the mean value of sales over the entire period considered. The average sales per month during a specified period can be crucial for determining consistent performance over time.

For annual performance, the average sales over 12 months are calculated by dividing the total sales by 12. Mathematically, this is represented as:

• \( \text{Average Sales per Month} = \frac{1}{12} \int_{0}^{12} r(t) \, dt \)

Calculating the average helps in assessing overall performance in a simple number that is easy to interpret and communicate. It provides a baseline that can be used for future comparisons and evaluations.

Additionally, knowing the average sales aids in budgeting and planning, ensuring that resources are correctly allocated to sustain and possibly enhance sales operations. If the average is below expectations, it could prompt a review of marketing or sales strategies to boost performance.