Related RatesSOL APC.8c

Luke Robbins, Sara Lasker, Michelle Bousquet

Steps to Solve any Related Rates Problem

1) Draw and label a diagram to visually represent the problem.

2) Define the variables.3) List the givens and the unknown(s).4) Brainstorm possible geometric or algebraic relationships

between the variables and choose the relationship that contains all the givens and the unknown(s).

5) Differentiate the equation implicitly with respect to time.

6) Solve for the unknown variable(s).7) Interpret the solution in the context of the problem.

The Problem

A 20-foot long ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of 10 , how fast is the top of the ladder sliding down the wall when the top of the ladder is 5 feet from the ground?

This diagram represents the ladder leaning against a wall. The wall has a 90 degree angle with the ground.

Step 1) Draw and label a diagram to visually represent the problem.

10 feet/second

20 feet

5 feet

Variablesx = distance from the wall to the bottom of the laddery = distance from the top of the ladder to the floorz = length of ladder (constant)t = time (in seconds)rate at which x is increasingrate at which y is increasing

Step 2) Define the variables.

x

yz = 20 feet

Givensy = 5 feetz = 20 feet10

Step 3) List the givens and the unknown(s).

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

x = ?

y = 5 feetz = 20 feet

10

?

Step 4) Brainstorm possible geometric or algebraic relationships between the variables and choose the relationship that contains all the givens and the unknown.

Possible equationsA =

x = ?

y = 5 feetz = 20 feet

10

?

Step 4) Brainstorm possible geometric or algebraic relationships between the variables and choose the relationship that contains all the givens and the unknown.

Possible equationsA =

We choose this relationship because it and its derivative include all givens and the unknown.

Givensy = 5 feetz = 20 feet10

5) Differentiate the equation implicitly with respect to time.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

Geometric Relationship

Givensy = 5 feetz = 20 feet10

5) Differentiate the equation implicitly with respect to time.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

We know that z is constant, so we can plug in the value of z.

Givensy = 5 feetz = 20 feet10

5) Differentiate the equation implicitly with respect to time.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

We derive the equation with respect to time, t.

Givensy = 5 feetz = 20 feet10

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

At this point, we have two unknowns. Luckily we can calculate x with the original equation.

Givensy = 5 feetz = 20 feet10

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

We will isolate x in the original equation as an intermediate solution.

Givensy = 5 feetz = 20 feet10

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ??

10

?

We substitute in y to find what x is when y=5 feet.

Givensy = 5 feetz = 20 feet10

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

Unknownsx = ?

10

?

feetNow we have all the givens to solve for

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

Now we solve this equation for

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

6) Solve for the unknown variable.

x = ?

y = 5 feetz = 20 feet

10

?

7) Interpret the solution in the context of the problem.

x = ?

y = 5 feetz = 20 feet

10

?

The ladder is moving down the wall at a rate of 38.73 feet per second.