### All Calculus 1 Resources

## Example Questions

### Example Question #1 : Rate

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

**Possible Answers:**

liters per second

liters per second

liters per second

liters per second

**Correct answer:**

liters per second

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

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

If water is poured inside a cylindrical cup with radius of 3 inches at a rate of 2 inches per second, how fast is the surface rising?

**Possible Answers:**

**Correct answer:**

Write the volume of the cylinder.

Substitute the radius.

Differentiate the volume with respect to time.

Substitute the given rate.

The height is increasing at a rate of inches per second.

### Example Question #3 : Rate

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?

**Possible Answers:**

**Correct answer:**

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 #4 : How To Find 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?

**Possible Answers:**

**Correct answer:**

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 #1 : 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)?

**Possible Answers:**

**Correct answer:**

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 cm^{3}/second. Plugging in our known values, we can thus solve for dh:

### Example Question #5 : Rate

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?

**Possible Answers:**

**Correct answer:**

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 #2 : 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?

**Possible Answers:**

**Correct answer:**

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 #4 : How To Find 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?

**Possible Answers:**

**Correct answer:**

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 #8 : Rate

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?

**Possible Answers:**

units per second

units per second

units per second

units per second

units per second

**Correct answer:**

units per second

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 #9 : Rate

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?

**Possible Answers:**

**Correct answer:**

Given,

and the power rule

where , then

.

Therefore, at ,

liters per minute.

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