Problem 44 - Applied Calculus

Question

In an effort to make room for new inventory, a college bookstore runs a sale on its least popular mathematics books. The sales rate (books sold per day) on day \(t\) of the sale is predicted to be \(60 / t\) (for \(t \geq 1\) ), where \(t=1\) corresponds to the beginning of the sale, at which time none of the inventory of 350 books had been sold. a. Find a formula for the number of books sold up to day \(t\) b. Will the store have sold its inventory of 350 books by day \(t=30 ?\)

Step-by-Step Solution

Answer

The store will not have sold all 350 books by day 30; only 264 books are predicted to be sold by then.

Step 1: Understand the Problem

Identify what is given and what is asked. We know the sales rate is given by \(60/t\), and initially, no books are sold. We need to find the total number of books sold by day \(t\), and check if all 350 books will be sold by day \(t=30\).

Step 2: Set Up the Integral for Sales Function

The rate of sales is a continuous function, \(r(t) = \frac{60}{t}\). The number of books sold up to day \(t\) can be found by integrating this rate from 1 to \(t\). Thus, the number of books sold on or before day \(t\), \(N(t)\), is given by the integral:\[N(t) = \int_{1}^{t} \frac{60}{x} \, dx\]

Step 3: Compute the Integral

Evaluate the integral \(\int \frac{60}{x} \, dx\) to find the expression for \(N(t)\). The integral of \(\frac{60}{x}\) gives:\[N(t) = 60 + 60 \ln(t)\]

Step 4: Verify the Inventory Sold by a Specific Day

Calculate \(N(30)\) using the function \(N(t) = 60 + 60 \ln(t)\) and compare it to 350. Compute\[N(30) = 60 + 60 \ln(30)\] and determine if this is greater than or equal to 350.

Step 5: Calculate \(\ln(30)\) and Solve

The approximate value of natural logarithms is needed. Using \(\ln(30) \approx 3.4\), we find:\[N(30) = 60 + 60 \times 3.4 = 60 + 204 = 264\]

Step 6: Conclude Based on Calculations

Since \(N(30) = 264\), which is less than 350, the store will not have sold all 350 books by day 30.

Key Concepts

Sales Rate Function

In this context, the sales rate function is a mathematical expression that estimates how quickly books are being sold over time.
For this exercise, the sales rate function is given as \( r(t) = \frac{60}{t} \), where \( t \) represents the day number starting from 1.
Essentially, this function tells us that as each day of the sale passes, the rate at which books are sold decreases.

Sales rate functions are crucial in predicting future sales trends and are often utilized in various fields such as inventory management and marketing. Understanding this can help businesses optimize their sales strategies and manage resources more efficiently.

Definite Integral

The concept of a definite integral is fundamental in solving calculus word problems like this one.
When we talk about a definite integral, we're referring to the exact calculation of an accumulation—in this case, the total number of books sold over a period of time.
To find out how many books have been sold by a certain day \( t \), we integrate the sales rate function \( r(t) = \frac{60}{t} \) from day 1 to day \( t \), expressed mathematically as:

• \[ N(t) = \int_{1}^{t} \frac{60}{x} \, dx \]

In practical terms, the definite integral tells us the area under the sales rate curve between two points, which equates to the total sales up to that point in time.
Evaluating these integrals gives us insights into cumulative data, crucial for real-world applications.

Bookstore Inventory Problem

This classic inventory problem involves understanding how long it will take to sell off an existing stock using a given sales rate.
Inventory management is critical for businesses to avoid overstocking or running out of items.

In this exercise, the bookstore's initial inventory is 350 books of a certain type. The question is whether this entire inventory will be sold by the 30th day.
The solution requires first finding the expression for the total number of books sold by day \( t \), \( N(t) = 60 + 60 \ln(t) \), based on the integral of the sales rate.
By calculating \( N(30) \) using this sales function, we found it to be 264, confirming that at the end of 30 days, the bookstore will have sold 264 books.
Thus, the inventory is not entirely sold by the end of that period.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), appears frequently in calculus word problems and plays a significant role in calculating integrals of certain functions.
In this scenario, the integral of the sales rate \( \frac{60}{x} \) resulted in an expression that includes \( \ln(t) \).

So what exactly is a natural logarithm? It is a logarithm to the base \( e \), where \( e \approx 2.71828 \), a fundamental constant in mathematics.
The function \( \ln(x) \) is particularly useful for handling exponential relationships and is integral in various calculus operations.

• For instance, the logarithmic integral of \( \frac{1}{x} \) is explained using natural logarithms.

In this exercise, you calculate \( \ln(30) \) ≈ 3.4 to determine how many books have been sold by the 30th day.
This understanding helps in accurately modeling and resolving real-world problems involving rates and growth.