# Calculus 1 : Rate of Flow

## Example Questions

### Example Question #1 : How To Find Rate Of Flow

The volume  in liters of water in a tank at time   in seconds is .  What is the rate of flow from the tank at

liters per second

liters per second

liters per second

liters per second

liters per second

Explanation:

To find the rate of flow, you need to differentiate the function.  This is  substituting  in for  gives the answser of .

### Example Question #2 : How To Find Rate Of Flow

Suppose a fish tank has a shape of a square prism with a length of 10 inches. If a hose filled the tank at 3 cubic inches per second, how fast is the water's surface rising?

Explanation:

Given the length of the cube is 10 inches, the length and the width is also 10 inches.  However, the height of the water is unknown.  Let's assume this height is .

Write the volume of the water in terms of .

Differentiate the volume equation with respect to time.

Substitute the rate of the water flow into .

The water is rising at a rate of  inches per second.

### Example Question #1 : Rate Of Flow

Make sure you identify what this question is asking.

A large vat contains  of butter. The vat has a small leak, out of which,   of butter escapes every hour. What is the rate of change in the volume of butter in the vat?

Explanation:

This question tells you that there is a leak of  mL/hour, and then asks you to identify how quickly the leak is causing butter to be lost.

The key is to identify the units, mL/hour, see that there are  mL/hour being moved, and recognize that the units are negative, as  mL are being removed every hour.

Thus the rate of change in the volume of the butter is .

### Example Question #4 : How To Find Rate Of Flow

A cylindrical tank with a radius of 20 centimeters and an arbitrary height is filling with water at a rate of 1.5 liters per second.  What is the rate of change of the water level (its height)?

Explanation:

The first step we may take is to write out the formula for the volume of a cylinder:

Where r represents the radius and h the height.

With this known we can find the rate change of volume with respect to height, by deriving these functions with respect to height:

Since we're interested in the rate change of height, the dh term, let's isolate that on one side of the equation:

dV, the rate change of volume, is given to us as 1.5 liters per second, or 1500 cm3/second.  Plugging in our known values, we can thus solve for dh:

### Example Question #4 : How To Find Rate Of Flow

The volume  of water (in liters) in a pool at time  (in minutes) is defined by the equation  If Paul were to siphon water from the pool using an industrial-strength hose, what would be the rate of flow at  in liters per minute?

Explanation:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given

, we can apply the power rule,

Then

Therefore, at

liters per minute.

### Example Question #3 : How To Find Rate Of Flow

The volume  of water (in liters) in a river at time  (in minutes) is defined by the equation . What is the river’s rate of flow at  in liters per minute?

Explanation:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given,

, we can apply the power rule,

to find

.

Therefore, at

liters per minute.

### Example Question #2 : Rate Of Flow

The volume  of water (in liters) in a tank at time  (in minutes) is defined by the equation . What is the tank’s rate of flow at  in liters per minute?

Explanation:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given,

, then using the power rule which states,

thus .

Therefore, at ,

liters per minute.

### Example Question #2 : How To Find Rate Of Flow

The area bouned by the curve  and  is being filled up with water at a rate of  unit-squared per second. When the water-level is at  how quickly is the water-level rising?

units per second

units per second

units per second

units per second

units per second

units per second

Explanation:

If , then . Therefore the width of the water at this level is . Some small change  in height is associated with a change of  in area. In other words:

Taking the reciprocal yields:

We also know that  is increasing at 2 unit-squared per second, so:

Therefore:

Therefore, the height will be rising at  units per second.

### Example Question #5 : How To Find Rate Of Flow

The volume  of water (in liters) in a stream at time  (in minutes) is defined by the equation . What is the stream’s rate of flow at  in liters per minute?

Explanation:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given,

and the power rule

where ,  then

Therefore, at ,

liters per minute.

### Example Question #6 : How To Find Rate Of Flow

The volume  of water (in liters) in a pool at time  (in minutes) is defined by the equation . What is the pool’s rate of flow at   in liters per minute?

Explanation:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given

and the power rule

where , then

.

Therefore, at

liters per minute.