Problem 79 - Applied Calculus

Question

After \(t\) hours of work, a bank clerk can process checks at the rate of \(r(t)\) checks per hour for the function \(r(t)\) given below. How many checks will the clerk process during the first three hours (time 0 to time 3 )? $$ r(t)=-t^{2}+90 t+5 $$

Step-by-Step Solution

Answer

The bank clerk will process 411 checks during the first three hours.

Step 1: Set Up the Integral

To find the total number of checks processed from time 0 to time 3, we need to integrate the rate function \(r(t)\) from \(t = 0\) to \(t = 3\). This integral will give us the accumulated checks processed over the time period. The integral is \( \int_{0}^{3} (-t^2 + 90t + 5) \, dt \).

Step 2: Integrate the Function

We will integrate each term in the function \(-t^2 + 90t + 5\) separately. The integral for each term is:- \( \int -t^2 \, dt = \frac{-t^3}{3} + C \)- \( \int 90t \, dt = 45t^2 + C \)- \( \int 5 \, dt = 5t + C \)Combining these, the indefinite integral is: \( F(t) = \frac{-t^3}{3} + 45t^2 + 5t + C \).

Step 3: Compute the Definite Integral

Evaluate the definite integral from \(t = 0\) to \(t = 3\):- \( F(3) = \frac{-(3)^3}{3} + 45(3)^2 + 5(3) \)- \( F(0) = \frac{-0^3}{3} + 45(0)^2 + 5(0) \)Calculate these values:- \( F(3) = \frac{-27}{3} + 405 + 15 = -9 + 405 + 15 \)- \( F(3) = 411 \)- \( F(0) = 0 \)Subtract \(F(0)\) from \(F(3)\) to find the total checks processed: - Total = 411 - 0 = 411.

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics that helps us understand accumulation and total change over time. In this exercise, we're working to find out how many checks the bank clerk processes over a specific time frame. We achieve this by using a definite integral of the given function, which represents the rate at which the clerk processes checks.

This approach involves two core activities:

• Establishing the integral of the rate function to derive a mathematical expression that represents the total checks processed.
• Calculating the definite integral over the specified interval — here, from 0 to 3 hours.

By setting up the integral as \( \int_{0}^{3} (-t^2 + 90t + 5) \, dt \), we can compute the total checks processed. This is a practical application of integral calculus, highlighting how continuous data (such as a varying rate over time) can be summed up to give a total quantity (checks in this case). Understanding the mechanics of integrating a function is vital, as this paves the way to solve many real-world problems.

Applications of Integration

Using integrals, we can handle problems that involve accumulation, which is significant in areas like physics, economics, and everyday situations. In this task, integration is used to model the number of checks processed by determining the total output during a certain time period.

This technique becomes useful for:

• Calculating the total change or quantity over a certain interval, such as determining total sales, or in our case, processed checks.
• Summing up or accumulating varying rates of change — like figuring out how much quantity was produced over a specific timeframe when the rate isn't constant.

Such applications help visualize how independent variables affect dependent results over time. By integrating the rate function, we calculate the total checks processed from start to end of the given interval. This highlights the potential of integration beyond pure mathematics, making it invaluable for solving practical issues.

Rate of Change

The concept of rate of change is common in various fields like physics and economics, representing how one quantity alters in response to another. In this case, the rate function \( r(t) = -t^2 + 90t + 5 \) specifically tells us how many checks are processed per hour. Here, integral calculus steps in to address how the total amount accumulates over time.

By understanding rate of change:

• We can evaluate instantaneous changes and predict trends.
• We're able to integrate these rates to arrive at total values, such as the checks processed over the first three hours.

This focus on the rate provides insight into the process dynamics, showing acceleration (or deceleration) and volume throughput. Realizing these patterns via integration allows us to predict future behavior or gain knowledge into past performance. In practical terms, mastering the link between rates and totals empowers decision-making and planning by dissecting rate variations and their impact over time.