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§3.4 Related Rates.

The student will learn about

related rates.

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Introduction to Related Rates

All of our relationships involve two variables. We have found the derivative with respect to one of those variables in order to find a rate of change in that variable.

Related rate problems find the rate of change of both variables (implicitly) with respect to time and solve for one rate with respect to the other.

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Example 1A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high?

Consider making a drawing.

Note from the drawing that y and z are the variables.

What is an equation that relates y and z?

z 2 = y 2 + 300 2

y

300

z

dz/dt is the unknown.

dy/dt is given as 5 meters per second.

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Example 1 - continuedA weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? z 2 = y 2 + 300 2

Implicitly differentiate the equation.

and solving for dz/dt yields,

and300dt

dy

dt

dz

dt

d 22

0dt

dyy2

dt

dzz2

dt

dy

z

y

dt

dy

z2

y2

dt

dz

y

300

z

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Example 1 - continuedA weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high?

And we know dy/dt = 5 meters/sec.When y = 400, z = 500. Why?

dt

dy

z

y

dt

dz

.sec/meter45500

400

dt

dz

z 2 = 400 2 + 300 2

y

300

z

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Example 2A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? [Use A = π R 2 ]

Making a drawing.Note dr/dt = 2 and R = 10 are given. Find dA/dt.

Substitute what we know.

R

dt

dRR2

dt

dA

402102dt

dA

A = π R 2

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Solving Related Rate ProblemsStep 1. Identify the quantities that are changing with time. Express these rates as derivatives.

Step 2. Identify all variables, including those that are given and those to be found.

Step 3. Find an equation connecting variables. Sketch a figure if helpful.

Step 4. Implicitly differentiate this equation.

Step 5. Substitute into the new equation any given values for the variables and the derivatives.Step 6. Solve for the remaining derivative and interpret the answer as the unknown rate.

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Example 3The sides of a cube are increasing at 3 cm per second. How fast is the volume increasing when the edges are 5 cm?

Making a drawing.

V = volume of cube = 5 3 = 125

ds cm3 sec .dts = side of the cube = 5

s = 6

Step 1. Identify the quantities that are changing with time. Express these rates as derivatives.

ds3cm sec .

dt

Step 2. Identify all variables, including those that are given and those to be found.

dVunknown

dt

dVunknown

dt

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Example 3The sides of a cube are increasing at 3 cm per second. How fast is the volume increasing when the edges are 6 cm?

ds cm3 sec .dts = 5 V = 125

s = 6 2dV ds

3sdt dt

Step 3. Find an equation connecting variables.

V = s 3

Step 4. Implicitly differentiate this equation.

Step 5. Substitute into the new equation any given values for the variables and the derivatives.

32dV cm3 5 3 225 secdt

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Example 3The sides of a cube are increasing at 3 cm per second. How fast is the volume increasing when the edges are 6 cm?

ds cm3 sec .dts = 5 V = 125

s = 6

Notice you do not need V in the problem.

32dV cm3 5 3 225 secdt

Step 6. Solve for the remaining derivative and interpret the answer as the unknown rate.

Did you get the units correct?

Do you understand the 225 cm 3 / second?

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Summary.

• We learned that related rate problems occur in life and in business and they need calculus for their solutions.

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ASSIGNMENT

§3.5; Page 65; 1 - 9, odd.