Practice Problem Set: Rate-In/Rate-Out Problems

The Core Idea

Rate-in/rate-out problems describe a quantity that changes because something is entering and/or leaving a system. Common examples include water flowing into a tank, people entering and leaving a stadium, medication entering and leaving the bloodstream, or pollutants entering and leaving a reservoir.

The key idea is simple: what is left at the end equals what you started with plus everything that entered minus everything that left.

When the rates are functions of time, integration gives the total amount accumulated over an interval.

How to Solve Rate-In/Rate-Out Problems

Identify the quantity. Decide what is being measured: gallons, people, kilograms, milligrams, and so on.
Identify the initial amount. This is your starting value, such as .
Identify the rate in and rate out. Both should have units of quantity per unit time.
Find the net rate. Subtract the rate out from the rate in.
Integrate the net rate over the appropriate time interval.
Add the resulting net change to the initial amount.
Check units and reasonableness. Your final answer should have units of the original quantity.

Worked Examples

Example 1: Water Tank

A tank initially contains gallons of water. Water enters the tank at a rate of gallons per minute and leaves at a rate of gallons per minute, where . How much water is in the tank at ?

Step 1: Net rate = rate in − rate out:

Step 2: Net change from to :

Step 3: Add the initial amount:

Result:

Example 2: Population of a Stadium

A stadium has people at noon. People enter at a rate of people per minute and leave at a rate of people per minute. Find the number of people in the stadium at .

Step 1: The net rate of change is .

Step 2: The accumulated net change is .

Step 3: Add the initial population: .

Result:

Example 3: When the Rate-In and Rate-Out Functions Are Given

A reservoir contains cubic meters of water at . Water flows in at rate and flows out at rate , where . Determine the amount of water at .

Step 1: Net rate:

Step 2: Net change:

Step 3: Evaluate the integrals:

Result:

Example 4: Finding When the Quantity Is Increasing

A tank has an inflow rate and an outflow rate . At what times is the amount of water in the tank increasing?

Step 1: The amount increases whenever the net rate is positive: .

Step 2: Solve the inequality .

Step 3: On the specified interval, this gives .

Interpretation: The tank gains water whenever the inflow exceeds the outflow.

Common Mistakes

Adding the outflow instead of subtracting it

Outflow decreases the quantity. The correct net rate is , not .

Forgetting the initial amount

The integral gives the net change, not automatically the final amount. If the system starts with , you must add that starting amount to the accumulated change.

Using the wrong time interval

If the problem asks for the amount at and gives an initial amount at , integrate from the initial time to the final time.

Confusing rate with amount

A rate such as gallons per minute is not an amount of water. Integration converts the rate into an accumulated quantity.

Practice Problems

Question 1

A tank initially contains liters of water. Water enters at liters per minute and leaves at liters per minute for minutes. How much water is in the tank after 10 minutes?

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Question 2

A reservoir contains cubic meters of water at time . The rate of inflow is cubic meters per hour, while the rate of outflow is cubic meters per hour. Find the amount of water at .

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Question 3

People enter a museum at rate and leave at rate , where . If 120 people are inside at 9:00 AM, how many people are inside at 11:00 AM?

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Question 4

A tank initially contains gallons. Its rate of change is gallons per minute. How much does the amount of water change from to ?

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Question 5 (Increasing or Decreasing)

Water enters a tank at rate and leaves at rate . On what intervals is the amount of water increasing?

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Question 6 (AP-Style)

A tank contains gallons at time . Water enters at rate gallons per hour and leaves at rate gallons per hour. Estimate the amount of water in the tank at using the trapezoidal rule with four equal subintervals.

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A Reliable AP Calculus Strategy

When you see a rate-in/rate-out problem, immediately write the structure before doing any calculations:

Then ask three questions:

What is the starting amount?
What is the net rate? Inflow minus outflow.
What interval am I accumulating over?

This prevents most sign errors. On an AP Calculus free-response question, also include units and an interpretation of your answer when requested.

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