Related Rates SOL APC.8c

1. Related RatesSOL APC.8c Luke Robbins, Sara Lasker, Michelle Bousquet

2. Steps to Solve any Related Rates Problem • Draw and label a diagram to visually represent the problem. • Define the variables. • List the givens and the unknown(s). • Brainstorm possible geometric or algebraic relationships between the variables and choose the relationship that contains all the givens and the unknown(s). • Differentiate the equation implicitly with respect to time. • Solve for the unknown variable(s). • Interpret the solution in the context of the problem.

3. 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?

4. Step 1) Draw and label a diagram to visually represent the problem. This diagram represents the ladder leaning against a wall. The wall has a 90 degree angle with the ground. 20 feet 5 feet 10 feet/second

5. Step 2) Define the variables. Variables x = distance from the wall to the bottom of the ladder y = distance from the top of the ladder to the floor z = length of ladder (constant) t = time (in seconds) rate at which x is increasing rate at which y is increasing z = 20 feet y x

6. Step 3) List the givens and the unknown(s). Unknowns x = ? ? Givens y = 5 feet z = 20 feet 10 ? z = 20 feet y = 5 feet x = ? 10

7. Step 4) Brainstorm possible geometric or algebraic relationships between the variables and choose the relationship that contains all the givens and the unknown. Possible equations A = ? z = 20 feet y = 5 feet x = ? 10

8. Step 4) Brainstorm possible geometric or algebraic relationships between the variables and choose the relationship that contains all the givens and the unknown. Possible equations A = We choose this relationship because it and its derivative include all givens and the unknown. ? z = 20 feet y = 5 feet x = ? 10

9. 5) Differentiate the equation implicitly with respect to time. Geometric Relationship ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

10. 5) Differentiate the equation implicitly with respect to time. We know that z is constant, so we can plug in the value of z. ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

11. 5) Differentiate the equation implicitly with respect to time. We derive the equation with respect to time, t. ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

12. 6) Solve for the unknown variable. At this point, we have two unknowns. Luckily we can calculate x with the original equation. ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

13. 6) Solve for the unknown variable. We will isolate x in the original equation as an intermediate solution. ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

14. 6) Solve for the unknown variable. We substitute in y to find what x is when y=5 feet. ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? ? y = 5 feet x = ? 10

15. 6) Solve for the unknown variable. feet Now we have all the givens to solve for ? Givens y = 5 feet z = 20 feet 10 z = 20 feet Unknowns x = ? y = 5 feet x = ? 10

16. 6) Solve for the unknown variable. Now we solve this equation for ? z = 20 feet y = 5 feet x = ? 10

17. 6) Solve for the unknown variable. ? z = 20 feet y = 5 feet x = ? 10

18. 6) Solve for the unknown variable. ? z = 20 feet y = 5 feet x = ? 10

19. 6) Solve for the unknown variable. ? z = 20 feet y = 5 feet x = ? 10

20. 6) Solve for the unknown variable. ? z = 20 feet y = 5 feet x = ? 10

21. 6) Solve for the unknown variable. ? z = 20 feet y = 5 feet x = ? 10

22. 7) Interpret the solution in the context of the problem. The ladder is moving down the wall at a rate of 38.73 feet per second. ? z = 20 feet y = 5 feet x = ? 10