# Calculus 1 : How to find rate of flow

## Example Questions

### Example Question #31 : How To Find Rate Of Flow

Water is spilling from a cone at .  The cone has a radius of  at the top and is  high.  At what rate is the depth of the water changing when the water reaches a height of ?

Explanation:

We will use the equation for the volume of a cone and its derivative to solve this problem.

The equation for the volume of a cone is

where  is the radius of the cone at the top and  is the height of the cone.

Since we do not know , we must eliminate  from the volume equation before we differentiate.

We use the ratio of height to radius to eliminate .

The volume equation is now

We will now take the derivative of the equation with respect to time or .  We will use the power and chain rules to find the derivative.

We are given the following parameters

and we need to find  at .

Subsituting this information into the first derivative

### Example Question #32 : How To Find Rate Of Flow

Water is poured into a cone shaped cup at a rate of 1 m/s. The radius of the cone is 5 m. What is the rate of change of the height of the water in the cone?

Explanation:

The equation for the volume of a cone is . To find the rate of change of height we must differentiate with respect to h and treat r as a constant.

We can plug in the flow rate of 1 and solve for the rate of change of height.

### Example Question #33 : How To Find Rate Of Flow

A cylindrical water tank has a 2 feet radius and is 5 feet long.  If the tank is halfway full of water that weighs 25 pounds per cubic feet, what is the force by the water acting on one of the tank's ends?

Explanation:

For this problem, the length of the tank is irrelevant.

Write the formula for the force exerted by a fluid.

We will need to find the equation of the of circle given the radius.

The equation of a circle is:

The center of the circle is .

Rewrite the equation in terms of .

The length of the horizontal rectangular strips are twice the length of :

The height , or the depth, is  assuming that the downward direction of the y-axis is positive.

The pressure  is .

The bounds are from the center of the circle to the outer rim of the tank, which is a distance of two.  The bounds are from zero to two.  Write the integral and solve.

Use substitution to solve.

Replace the variables of  to in terms of  in order to solve the integral.  Pull out all constants out in front of the integral.

Re-substitute .

### Example Question #34 : How To Find Rate Of Flow

A cone-shaped funnel with a base diameter of  and a height of  is initially full of water, though water begins to drain from the tip at a rate of  . How fast does the water level fall when it's at a height of  ?

Explanation:

It might be assumed intuitively that the water level will decrease faster as it lowers towards the tip of the cone. To model this, relate the radius of the plane of the water to the height of the water:

Now, since the volume of water at a point in time is approximately a cone shape, it can be written as:

To find the rate of change, take the derivative with respect to time:

The term  is unknown and is what we're solving for, but we're told that .

At height :

### Example Question #31 : How To Find Rate Of Flow

A cone-shaped funnel with a base diameter of  and a height of  is initially full of water, though water begins to drain from the tip at a rate of  . How fast does the radius of the water's surface diminish when it's at a height of ?

Explanation:

Itmay be guessed that the radius of the water's surface will decrease faster as it lowers towards the tip of the cone, i.e it's not constant. To model this, relate the radius of the plane of the water to the height of the water:

Now, since the volume of water at a point in time is approximately a cone shape, it can be written as:

To find the rate of change, take the derivative with respect to time:

The term  is unknown and is what we're solving for, but we're told that .

At height :

### Example Question #36 : How To Find Rate Of Flow

A cone-shaped funnel with a base radius of  and a height of  is initially full of water, though water begins to drain from the tip at a rate of  . How fast does the water level fall when it's at a height of  ?

Explanation:

It might be assumed intuitively that the water level will decrease faster as it lowers towards the tip of the cone.

To model this, relate the radius of the plane of the water to the height of the water:

Now, since the volume of water at a point in time is approximately a cone shape, it can be written as:

To find the rate of change, take the derivative with respect to time:

The term  is unknown and is what we're solving for, but we're told that .

At height :

### Example Question #37 : How To Find Rate Of Flow

A cone-shaped funnel with a base circumference of  and a height of  is initially full of water, though water begins to drain from the tip at a rate of  . How fast does the water level fall when it's at a height of  ?

Explanation:

It might be assumed intuitively that the water level will decrease faster as it lowers towards the tip of the cone.

To model this, relate the radius of the plane of the water to the height of the water:

Now, since the volume of water at a point in time is approximately a cone shape, it can be written as:

To find the rate of change, take the derivative with respect to time:

The term  is unknown and is what we're solving for, but we're told that .

At height :

### Example Question #38 : How To Find Rate Of Flow

A cone-shaped funnel with a base diameter of  and a height of  is initially full of water, though water begins to drain from the tip at a rate of  . How fast does the water level fall when it's at a height of  ?

Explanation:

It might be assumed intuitively that the water level will decrease faster as it lowers towards the tip of the cone.

To model this, relate the radius of the plane of the water to the height of the water:

Now, since the volume of water at a point in time is approximately a cone shape, it can be written as:

To find the rate of change, take the derivative with respect to time:

The term  is unknown and is what we're solving for, but we're told that .

At height :

### Example Question #1831 : Functions

Water is being added to an empty pool using a hose that sprays out water at a constant rate of flow. The pool holds 500 gallons of water, and took 3 hours to finish filling up.

Find an expression for the total amount of water in the pool between t = 0 and t = 180 minutes.

(Hint: Find the rate in which the water was flowing out of the hose and into the pool)

Explanation:

The rate of flow for the pool can be found by taking the total amount of water that flowed from the hose (the 500 gallons) divided by the amount of time it took to give those 500 gallons (3 hours or 180 minutes). Therefore, the rate of flow of the hose is determined to be 500/180 = 2.78 gallons/min. Afterwards, in order to develop an expression for the amount of water in the pool at a given time, you can take the integral of the flow. This integral will sum all the water that has been added up to that point and give you the total amount of water in the pool.

This makes sense because 2.78 gallons are being added every minutes, so the total number of minutes passed multipled by the 2.78 gallons gives the total amount of gallons in the pool. Notice that the C constant was not included, this is because the pool is originally empty. If the pool was not originally empty, the intial amount of water in the pool would have been set as the value of C.

### Example Question #40 : How To Find Rate Of Flow

Determine the rate of flow of a liquid at  given that it's flow function is determined by: