**Related Rates**

The purpose of using related rates is to find how factors of a situation affect each other as they change over time.

How it Works

To find these rates, we need to use implicit differentiation.

Example

The company Solution Union makes nontraditional ice sculptures. They were asked by a customer to make an ice Pyramid for an Egyptian themed party. The Pyramid has a height of 7 feet and a square base with sides of 8 feet. The ice melts at a rate of 3.45ft^2/minute. What is the rate at which the height is decreasing when the Pyramid has a volume of 312ft^3 and the area of it's base is 51.84? (The volume of a Pyramid is LWH/3)

Another Example

When it rains, drops of water in ponds create a ripple effect. These circular ripples have expanding circumferences. At what rate do are they expanding by the 7th ripple if the radius is expanding at a rate of 3 cm/sec and the circumference is 31.4cm?

Now You Try

Solution and Explanation

We know the circumference of the cone is 4π and that the height is 3 inches. We also know that the ice cream height is decreasing at a rate of .13in/minute and that you are looking for the rate at which the volume is decreasing.

First, plug in what you can get rid of and find any equations you need to get rid of variables.

1/3πhr^2=v

C=2πr, 4π=2πr, r=2

4/3πh=v

Now implicitly differentiate:

dv/dt=(4/3)πdh/dt

Solve

dv/dt= (4/3)π(-.13)

dv/dt=-.54454

This means that the volume of the cone is decreasing at a rate of -.5445 cubic inches per minute when the radius equals 2

Citations

Works Cited

Dawkins, Paul. "Pauls Online Notes : Calculus I - Related Rates." Pauls Online Notes : Calculus I - Related Rates. Paul Dawkins, 2015. Web. 20 May 2015.

"Formula Volume of a Triangular Prism." Formula Volume of Triangular Prism. Explained with Pictures and Examples. The Formula for ... Math Warehouse, n.d. Web. 20 May 2015.

"Khan Academy." Khan Academy. Khan Academy, 2015. Web. 20 May 2015.

Our Calculus Notebook

DON'T KILL THE PUPPY

(X+1)^2=...

The Purpose of Related Rates

First, look at what they give you and what you are looking for:

You know the dimensions of the pyramid and the rate at which it is melting. You are looking for the rate that the height is changing when the volume is 312ft^3 and the area of the base is 51.84 ft^2.

After you establish this, write a differentiated equation (and any other equations needed to get rid of unneeded variables):

L=W

(51.84)^(1/2)=7.2 312=51.84(H)/3 H=18.05

V=(W^2 *H)/3

W=(3V/H)^(1/2)

dW/dt=1/2 (936/18.05)^(-1/2) (3 (H *dV/dt- V * dH/dt)/H^2)

3.6* (3(18.05*3.45-312*dH/dt)/18.05^2)

70290778.64dH/dt

V=LWH/3 dV=1 2W dW H+W^2dH

dt 3 dt dt

Finally, you replace the values and alternate equations you have and solve:

3.45=1 14.4(18.05)(70290778.64dH/dt)+51.84dH/dt

3

3.45=6089993078dH/dt

5.66503107*10^-10=dH/dt

What does that number mean?

The height of the ice Pyramid is decreasing at a rate of 5.67*10^-10 ft/minute when the volume is 312ft^3.

Let's start another one!

We know that circles have a circumference of 2πr.

We also know that rate at which the radius is expanding, 3cm/second, and it's circumference, 31.4cm.

3.14=2πr r=.5

dc/dt=2πdr/dt

dc/dt=2π(3)

dc/dt=6π

The Circumference of the circles is increasing at a rate of 6π/second squared.

Max wanted to get chocolate ice cream at the Fair. The nearest ice cream stand was selling it in paper cones. The height of each cone is 3 inches. If Max licks away the ice cream so that the height is decreasing at a rate of .13inches/minute and the circumference of the of the cone is 4 π, at what rate is the volume decreasing (The volume of a cone is 1/3πhr^2)?

Remember how we start?

That's Right, we

implicit differentiation