# Calculus 1 : How to find rate of change

## Example Questions

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

A cylinder is growing wider and wider. If it has a radius of ten inches, a height of forty inches, and its radius expands at a rate of two inches per second, what is the rate of growth of its volume?

Explanation:

A cylinder's volume is defined in terms of its radius and height as:

Therefore, it's rate of volume expansion is found by taking the derivative of each side with respect to time:

Since there's zero growth in height, the second term goes to zero, and we're left with:

What an unnatural cylinder.

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

The surface area of a cone is given by the formula  where  is the radius of the cone at its base, and  is its height.

A cone with a radius of  and a height of  is widening, its radius growing at a rate of , though its height remains fixed. What is the rate of growth of its surface area?

Explanation:

To find the rate of growth of the area, take the time derivative of both sides of the surface area equation:

Since the height does not change with time, the fourth term is eliminated, leaving:

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

The volume of a hot hair balloon decreases at a rate of . What is the rate at which its radius decreases when the diameter of the balloon is ?

Explanation:

To solve this problem, first note the volume of the baloon as a function of its radius:

Now, derive both sides with respect to time, treating both volume and radius and functions of it:

The rate of change of the radius, , is what we are interested in, so rewrite the equation in terms of that:

The diameter of the baloon is , so its radius is  is given as , so the rate of change of the radius at this instant in time is:

### Example Question #2941 : Calculus

The sides of a square increase at a rate of . If the area of the square is , what is its rate of change?

Explanation:

Area of a square is given in terms of its sides as:

Deriving each side with respect of time allows us to find its rate of change:

The rate of change of the sides, , is given to us as .  To find the length of the side, use the orignal formula:

Therefore:

### Example Question #35 : How To Find Rate Of Change

A rectangle with an area of  and a perimeter of  is expanding. The longer side increases at a rate of  and the shorter side increases at a rate of . What is the rate of change of the rectangle's area?

Explanation:

Begin with finding the short and long sides of the rectangle, which we may designate as  and  respectively. Since the perimeter and area are given, we may write a system of equations:

Solvin for the two yields  and .

Now, to find the rate of change of the area, derive the area equation:

Since  and

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

A physician determines that the concentration of a medicine in the bloodsteam, in milligrams per liter, t minutes after it is given is modeled by the equation

.

At what rate is the concentration of the medicine in the bloodstream changing after 1 hour?

Explanation:

The derivative produces a new function that describes the rate of change of the original, so first calculate the derivative of c(t) using the product rule.

.

To find the rate of change when t=1, substitute 1 for t in the derivative.

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

A 13 foot ladder is leaning against the wall.  The bottom of the ladder is initially 10 feet away from the wall and being pushed toward the wall at a rate of .  How fast is the top of the ladder moving after 10 seconds?

Not enough information given.

Explanation:

In this question we are asked to find how fast the top of the ladder is going down.  To solve this quesiton, we must first realize that the ladder forms a triangle with the wall and the ground.

By realizing this we then simply take the derivative of the Pythagorean Theorem,

using the power rule,

.

The derivative of any constant is 0, therefore the derivative of the Pythagorean Theorem becomes

.

We must now find the values for  and  in order to solve for .  We know that , the distance of the bottom of the ladder to the wall is intially 10 feet and is being pushed toward the wall at a rate of  for 10 seconds, therefore

.

To solve for  we simply plug  back into the Pythagorean Theorem in the beginning, obtaining .  Solving for , we find that .  Plugging back into derivative of the Pythagorean Theorem, we know the values for  making it easy to solve for .  Note that  is negative in this equation because  is decreasing not increasing.

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

A cup with a shape of an inverted cone that is being filled up with water has a base radius of 3 inches and height 5 inches.  If the water is being poured into the cup at a rate of  find the rate at which the water level of the cup is rising when the water is 4 inches deep.

Not enough information is given.

Explanation:

To solve this equation, we must first know that the equation for the volume of a cone is

where  is the base radius of the cone and  is the height of the cone.

We are asked to find the rate of which the water level of the cup changes after the water height in the cone is equal to 4 inches, therefore we take the derivative of the volume equation with respect to time.

To take the derivative of the volume equation, we must use the power rule

as well as the product rule .

The derivative comes out to be

.

We know that

based off of the information given to us in the beginning; now we must just solve for .

Note that  different then expected due to the ratio of the cone.

This ratio is

or .

### Example Question #39 : How To Find Rate Of Change

A spherical snowball stuck into an oven and is melting at a rate such that its volume is decreasing by .  When the snowball has a radius of 10 centimeters, how fast is the diameter of the snowball changing?  Assume only the surface of the snowball is melting at a uniform rate.

Not enough information is given.

Explanation:

In order to solve this question, we must first know that the volume of a sphere is defined as

.

We are asked to find the rate of which the diameter of the snowball changes when it is melting.  Therefore we must take the derivative of the volume equation with respect to time.  To do this we will need to utilize the power rule

.

The derivative becomes

and since we know that , we simply plug the variables into the equation and solve for .

However we are asked to find the rate of which the diamater of the snowball changes, therefore we multiply the change of rate of the radius by 2 and obtain that the diameter changes at a rate of .

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

A car 40 miles east of point A starts driving north at a rate of 30 miles per hour.  What is the rate at which the distance between the car and point A is changing after one hour?

None of the above.

Explanation:

In order to solve this question, we must first see that a triangle is being formed between the car and point A. Therefore a Pythagorean theorem derivative can be used to solve this problem. The Pythagorean therorem

can be derived with respect to time in using the power rule

to obtain

.

We know from the information given to us that

,

therefore simply plugging in and solving for  will give us the answer to this problem.  However, we must still solve for .   can be solved by plugging in the values of  and  back into the Pythagorean theorem.  is equal to 30 miles because after one hour that is how far the car has traveled.

Now we simply plug everything into the derivative equation and solve for