Question

A business is prospering in such a way that its total (accumulated) profit after \(t\) years is \(1000 t^{2}\) dollars. (a) How much did the business make during the third year (between \(t=2\) and \(t=3) ?\) (b) What was its average rate of profit during the first half of the third year, between \(t=2\) and \(t=2.5 ?\) (The rate will be in dollars per year.) (c) What was its instantaneous rate of profit at \(t=2 ?\)

Short answer

(a) $5000, (b) $4500/year, (c) $4000/year at t=2.

Step-by-step solution

Understanding Accumulated Profit Function

The accumulated profit after time \(t\) is given by the function \(P(t) = 1000t^2\). To find how much profit the business makes over a specific year, we need to find the change in accumulated profit over that period.

Calculate Profit for the Third Year

To find the profit made during the third year, calculate the change in accumulated profit from the start to the end of that year. This is done by evaluating \(P(t)\) at \(t=3\) and \(t=2\), and then finding the difference: \[\text{Profit during third year} = P(3) - P(2) = 1000(3)^2 - 1000(2)^2 = 9000 - 4000 = 5000.\] The profit made during the third year is \$5000.

Average Rate of Profit During First Half of Third Year

The average rate of profit is the total profit made over a time interval divided by the length of the interval. For the first half of the third year, the interval is from \(t=2\) to \(t=2.5\). Calculate:\[\text{Profit from } t=2 ext{ to } t=2.5 = P(2.5) - P(2) = 1000(2.5)^2 - 1000(2)^2 = 6250 - 4000 = 2250.\]The time interval is \(0.5\) years, so the average rate is:\[\text{Average rate} = \frac{2250}{0.5} = 4500 ext{ dollars per year}.\]

Instantaneous Rate of Profit at t=2

The instantaneous rate of profit at a specific time is given by the derivative of the profit function \(P(t)\) evaluated at that time. The derivative of \(P(t) = 1000t^2\) is \(P'(t) = 2000t\). At \(t=2\), we find:\[P'(2) = 2000(2) = 4000.\]The instantaneous rate of profit at \(t=2\) is \$4000 per year.

Key concepts

Accumulated Profit Function

An accumulated profit function helps us understand how total profit evolves over time. It represents the entire profit collected from the start of the business up until a certain point, represented by a variable, usually time, like years. In our exercise, the accumulated profit function is given as \( P(t) = 1000t^2 \). This tells us how profit accumulates over time.

To determine how much profit is made in a specific year, you look at the accumulated profit at the start and end of that year.

• For instance, the profit made in the third year would be the accumulated profit at \( t = 3 \) minus the accumulated profit at \( t = 2 \).

This is important because even though the accumulated profit increases, the actual profit made in a specific period requires evaluating the change between two time points.

Average Rate of Profit

The average rate of profit gives us a sense of the profit made per unit of time over an interval. It's like finding the average speed if you know the distance traveled in a certain amount of time. In our exercise, the interval of interest is between \( t = 2 \) and \( t = 2.5 \), which is the first half of the third year.

• To calculate, find the profit made over the interval using the accumulated profit function and then divide it by the length of the time interval:
• First, calculate \( P(2.5) - P(2) \) to find profit change, resulting in \( 2250 \).
• Then, divide by the time interval, \( 0.5 \) years, achieving an average rate of \( 4500 \) dollars per year.

Understanding the average rate helps businesses understand periods of high efficiency and decision making in financial management.

Instantaneous Rate of Profit

The instantaneous rate of profit indicates how fast profit is growing at a specific moment. Unlike the average rate, which considers a broader interval, the instantaneous rate focuses precisely on a single instant, similar to checking the speedometer of a car at a specific moment.

To find the instantaneous rate at \( t = 2 \):

• Evaluate the derivative of the profit function \( P(t) \) at \( t = 2 \).
• For \( P(t) = 1000t^2 \), the derivative \( P'(t) = 2000t \), represents the change in profit per year.
• At \( t = 2 \), \( P'(2) = 4000 \) dollars per year, the instantaneous rate.

Being able to calculate the instantaneous rate helps businesses adjust their strategies at specific points for optimal performance.

Derivative of a Function

Derivatives are a fundamental tool in calculus that help us understand how one quantity changes in relation to another. In the context of profit, derivatives allow us to determine how profit changes with time.

• The derivative can give us insights into both average and instantaneous rates of change.
• For our problem, the function \( P(t) = 1000t^2 \) results in a derivative \( P'(t) = 2000t \), describing how the profit changes yearly.

It's like looking under the hood of the accumulated profits to see the mechanics of change. By finding and interpreting a derivative, businesses can forecast how their profits might grow and see how past changes occurred.