1. Section 2.6 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net
Related Rates
Section 2.6
Calculus AP/Dual, Revised ©2017
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§2.6: Related Rates

2. Definitions
Related rates are found when there are two or more variables that all depend on another variable, usually time
Two or more quantities change as time changes
Since the variables are related to each other, the rates at which they change (their derivatives) are also related
Real-life problems rarely involve just a single variable. Most are written in terms of multiple variables. Related rate problems are real-life situations based on equations defined by rates of change. We can differentiate these problems using IMPLICIT DIFFERENTATION.
Remember to replace one variable before differentiating.
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§2.6: Related Rates

3. Before We Start… 1) Distance Formula 𝒅= 𝒚 𝟐 − 𝒚 𝟏 𝟐 + 𝒙 𝟐 − 𝒙 𝟏 𝟐
𝒅= 𝒚 𝟐 − 𝒚 𝟏 𝟐 + 𝒙 𝟐 − 𝒙 𝟏 𝟐
2) Area of Triangle
𝑨= 𝒃𝒉 𝟐
3) Volume of a Sphere
𝑽= 𝟒 𝟑 𝛑 𝒓 𝟑
4) Pythagorean Theorem
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
5) Area of a Circle
𝑨=𝛑 𝒓 𝟐
6) Volume of a Cube
𝑽= 𝒔 𝟑
7) Circumference of a Circle
𝑪=𝟐𝛑𝒓
8) Surface Area of a Sphere
𝑺𝑨=𝟒𝛑 𝒓 𝟐
9) Volume of a Cylinder
𝑽=𝛑 𝒓 𝟐 𝒉
10) Volume of a Cone
𝑽= 𝟏 𝟑 𝛑 𝒓 𝟐 𝒉
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§2.6: Related Rates

4. Review Solve for 𝒅 𝒅𝒕 of the equation, 𝒙 𝟐 + 𝒚 𝟐 = 𝒅 𝟐
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§2.6: Related Rates

5. Review
Rewrite this in calculus terms: “The area of a circle is increasing at a rate of 6 square inches per second”
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6. Steps
Sketch and label the diagram and make sure the given rates are increasing (+) or decreasing (–)
Write the words “find” along with what you are finding out (for AP testing)
Write all given information
Write the equation which pertains to the problem (i.e. Pythagorean, Distance, Surface Area, etc…)
Differentiate with respect to time
Substitute all known values
Solve for the desired quantity
LABEL with appropriate units!
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§2.6: Related Rates

7. Example 1 Solve 𝒅𝒚 𝒅𝒕 at 𝒙=𝟏, given 𝒚= 𝒙 𝟐 +𝟑 and 𝒅𝒙 𝒅𝒕 =𝟐 when 𝒙=𝟏
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§2.6: Related Rates

8. Your Turn Solve 𝒅𝒚 𝒅𝒕 at 𝒙=𝟑, given 𝒚=𝟐 𝒙 𝟐 −𝟑𝒙 and 𝒅𝒙 𝒅𝒕 =𝟐
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§2.6: Related Rates

9. Example 2 Solve 𝒅𝑽 𝒅𝒕 of the volume of the cone of the equation.
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10. 2/16/2019 7:14 PM
§2.6: Related Rates

11. Example 2 Solve 𝒅𝑽 𝒅𝒕 of the volume of the cone. 2/16/2019 7:14 PM
§2.6: Related Rates

12. Example 2 Solve 𝒅𝑽 𝒅𝒕 of the volume of the cone.
Change in Radius in respects to Time
Change in Height in Respects to Time
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§2.6: Related Rates

13. Justifications Show the equation.
List the ‘given’ and identify the ‘find’
Show the differentiation
Show the substitution of plugging in the equation AFTER taking the differentiation
VANUT: Verb, Answer, Noun, Units, and Time
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§2.6: Related Rates

14. The area is changing at a rate of 8π ft2/sec when r is at 4 feet.
Example 3
A pebble is dropped into a calm pond, causing ripples in the shape of concentric circles. The radius of the outer ripple is increasing at a rate of 𝟏 𝒇𝒆𝒆𝒕/𝒔𝒆𝒄. When the radius is 𝟒 𝒇𝒕., find the rate at which the area disturbed water is changing.
The area is changing at a rate of 8π ft2/sec when r is at 4 feet.
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§2.6: Related Rates

15. Example 4
The waves in a pond are circular. The radius increases at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄. When the radius is 5 ft., at what rate is the area changing?
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§2.6: Related Rates

16. The rate of area is changing at 20π ft.2/sec when r = 5 feet.
Example 4
The waves in a pond are circular. The radius increases at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄. When the radius is 5 ft., at what rate is the area changing?
The rate of area is changing at 20π ft.2/sec when r = 5 feet.
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17. Example 5
Air is being pumped into a spherical balloon so that its volume increases at a rate of 𝟏𝟎𝟎 𝒄𝒎𝟑/𝒔. How fast is the radius of the balloon increasing when the diameter is 50 cm?
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18. Example 5
Air is being pumped into a spherical balloon so that its volume increases at a rate of 𝟏𝟎𝟎 𝒄𝒎𝟑/𝒔. How fast is the radius of the balloon increasing when the diameter is 50 cm?
The radius of the balloon is increasing at 1/(25π) cm./sec when d = 50 cm.
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19. The rate of radius is increasing at 1/π in. /sec when r = 1 in.
Your Turn
A balloon in the shape of a sphere is being blown up. The volume is increasing at the rate of 𝟒 𝒊𝒏𝟑/𝒔𝒆𝒄. At what rate is the radius increasing when the radius is exactly 1 in.?
The rate of radius is increasing at 1/π in. /sec when r = 1 in.
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20. Geometer’s Sketchpad
Cone
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21. Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
Given:
Height = 5 meters
Radius of Cone = 4 meters
Height of Cone = 16 meters
𝒅𝑽 𝒅𝒕 = 2 meters3
Find 𝒅𝒉 𝒅𝒕
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§2.6: Related Rates

22. Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
Similar Triangle Proportion
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§2.6: Related Rates

23. Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
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§2.6: Related Rates

24. Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
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25. Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
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26. Water is RISING at a rate of 32/(25π) meters/min when h = 5 meters
Example 6
Water is pouring into an inverted conical tank (right circular cone with height of 16 meters and base radius of 4 meters) at 2 cubic meters per minute. How fast is the water level rising when the water in the tank is 5 meters deep?
Water is RISING at a rate of 32/(25π) meters/min when h = 5 meters
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27. Example 7
Water runs out of an inverted conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the tank is 5 feet and the height of the tank is 10 feet. How fast is the water level sinking when the water is 4 feet deep?
Find 𝒅𝒉 𝒅𝒕 when 𝒉 = 4 feet
Given 𝒅𝑽 𝒅𝒕 = –2 ft3, 𝒉 = 4 feet
Similar Triangle Proportion
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§2.6: Related Rates

28. Example 7
Water runs out of an inverted conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the tank is 5 feet and the height of the tank is 10 feet. How fast is the water level sinking when the water is 4 feet deep?
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§2.6: Related Rates

29. Example 7
Water runs out of an inverted conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the tank is 5 feet and the height of the tank is 10 feet. How fast is the water level sinking when the water is 4 feet deep?
Find 𝒅𝒉 𝒅𝒕 when 𝒉 = 4 feet
Given 𝒅𝑽 𝒅𝒕 = –2 ft3, 𝒉 = 4 feet
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§2.6: Related Rates

30. Water is DECREASING at a rate of 1/(2π) feet/min when h = 4 feet
Example 7
Water runs out of an inverted conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the tank is 5 feet and the height of the tank is 10 feet. How fast is the water level sinking when the water is 4 feet deep?
Find 𝒅𝒉 𝒅𝒕 when 𝒉 = 4 feet
Given 𝒅𝑽 𝒅𝒕 = –2 ft3, 𝒉 = 4 feet
Water is DECREASING at a rate of 1/(2π) feet/min when h = 4 feet
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§2.6: Related Rates

31. Water is DECREASING at a rate of 1/(2π) ft/min when h = 8 feet
Your Turn
A tank filled with water is in the shape of an inverted cone 20 feet high with a circular base (on top) whose radius is 5 feet. Water is running out of the bottom of the tank at a constant rate of 𝟐 𝒇𝒕𝟑/𝒎𝒊𝒏. How fast is the water level falling when the water is 8 ft. deep?
Water is DECREASING at a rate of 1/(2π) ft/min when h = 8 feet
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§2.6: Related Rates

32. Example 8
A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a constant rate of 𝟎.𝟎𝟓 𝒊𝒏/𝒔𝒆𝒄. The volume is 𝟏𝟐𝟖𝝅 𝒊𝒏3. At what rate is the length, 𝒉, changing when the radius is exactly 1.8 inches?
The rate of area is INCREASING at a rate of in/sec when r = 1.8 inches.
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33. Geometric Sketchpad Falling ladder 2/16/2019 7:14 PM
§2.6: Related Rates

34. Example 9
A 20 ft. ladder leans against a vertical building. If the bottom of the ladder slides away from the building at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄, how fast is the ladder sliding down the building when the ladder is exactly 12 feet above the ground?
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§2.6: Related Rates

35. Example 9
A 20 ft. ladder leans against a vertical building. If the bottom of the ladder slides away from the building at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄, how fast is the ladder sliding down the building when the ladder is exactly 12 feet above the ground?
𝒅𝒚 𝒅𝒕 =?
𝒅𝒙 𝒅𝒕 =𝟐 𝒇𝒕. 𝒔𝒆𝒄
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§2.6: Related Rates

36. Example 9
A 20 ft. ladder leans against a vertical building. If the bottom of the ladder slides away from the building at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄, how fast is the ladder sliding down the building when the ladder is exactly 12 feet above the ground?
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37. Example 9
A 20 ft. ladder leans against a vertical building. If the bottom of the ladder slides away from the building at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄, how fast is the ladder sliding down the building when the ladder is exactly 12 feet above the ground?
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38. The ladder is FALLING at a rate of 8/3 ft./sec. when ladder is 12 ft.
Example 9
A 20 ft. ladder leans against a vertical building. If the bottom of the ladder slides away from the building at a rate of 𝟐 𝒇𝒕./𝒔𝒆𝒄, how fast is the ladder sliding down the building when the ladder is exactly 12 feet above the ground?
The ladder is FALLING at a rate of 8/3 ft./sec. when ladder is 12 ft.
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39. Your Turn
A ladder 10 feet long rest against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 𝟏 𝒇𝒕./𝒔𝒆𝒄, how fast is the top of the ladder sliding down at the wall when the bottom of the ladder is 6 ft. from the wall?
The ladder is FALLING at a rate of 3/4 ft./sec. when ladder is 6 ft. from the wall.
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40. Example 10
A balloon rises at a rate of 3 meters per sec from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
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§2.6: Related Rates

41. Example 10
A balloon rises at a rate of 3 meters per sec from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
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42. Example 10
A balloon rises at a rate of 3 meters per sec from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
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43. Example 10
A balloon rises at a rate of 3 meters per sec from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
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44. Example 11
In the right-triangle shown, the angle 𝜽 is increasing at a constant rate of 2 radians per hour. At what rate is the side length of 𝒙 increasing when 𝒙=𝟒 𝒇𝒕.?
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45. Your Turn
A balloon raises at the rate of 10 feet per second from a point on the ground 100 feet from an observer. Find the rate of change of the angle of elevation to the balloon from the observer when the balloon is 100 feet from the ground.
The angle of elevation is INCREASING at a rate of 1/20 rad./sec. when the balloon is 100 ft. from the ground.
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46. Example 12
A boy 5 feet tall walks at the rate of 4 feet per second directly away from a street light which is 20 feet above the street. (a) At what rate is the tip of his shadow changing? (b) At what rate is the length of his shadow changing?
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47. Example 12a
A boy 5 feet tall walks at the rate of 4 feet per second directly away from a street light which is 20 feet above the street. (a) At what rate is the tip of his shadow changing? (b) At what rate is the length of his shadow changing?
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48. Example 12a
A boy 5 feet tall walks at the rate of 4 feet per second directly away from a street light which is 20 feet above the street. (a) At what rate is the tip of his shadow changing? (b) At what rate is the length of his shadow changing?
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49. Example 12a
A boy 5 feet tall walks at the rate of 4 feet per second directly away from a street light which is 20 feet above the street. (a) At what rate is the tip of his shadow changing? (b) At what rate is the length of his shadow changing?
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50. Example 12b
A boy 5 feet tall walks at the rate of 4 feet per second directly away from a street light which is 20 feet above the street. (a) At what rate is the tip of his shadow changing? (b) At what rate is the length of his shadow changing?
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51. It is negative because the thief is running to the left
Example 13
A flood light is on the ground 45 meters from a building. A thief 2 meters tall, runs from the flood light towards the building at 6 meters/sec. How rapidly is the length of the shadow on the building changing when he is 15 meters from the building?
It is negative because the thief is running to the left
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52. Example 13
A flood light is on the ground 45 meters from a building. A thief 2 meters tall, runs from the flood light towards the building at 6 meters/sec. How rapidly is the length of the shadow on the building changing when he is 15 meters from the building?
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53. Example 13
A flood light is on the ground 45 meters from a building. A thief 2 meters tall, runs from the flood light towards the building at 6 meters/sec. How rapidly is the length of the shadow on the building changing when he is 15 meters from the building?
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§2.6: Related Rates

54. AP Multiple Choice Practice Question 1 (non-calculator)
When the height of a cylinder is 12 cm and radius is 4 cm, the circumference of the cylinder is increasing at a rate of 𝝅 𝟒 cm/min and the height of the cylinder is increasing four times faster than the radius. How fast is the volume of the cylinder changing? 𝑽=𝝅 𝒓 𝟐 𝒉 (A) 𝝅 𝟒 𝒄𝒎 𝟑 /𝒎𝒊𝒏 (B) 𝟒𝝅 𝒄𝒎 𝟑 /𝒎𝒊𝒏 (C) 𝟏𝟐𝝅 𝒄𝒎 𝟑 /𝒎𝒊𝒏 (D) 𝟐𝟎𝝅 𝒄𝒎 𝟑 /𝒎𝒊𝒏
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55. AP Multiple Choice Practice Question 1 (non-calculator)
When the height of a cylinder is 12 cm and radius is 4 cm, the circumference of the cylinder is increasing at a rate of 𝝅 𝟒 cm/min and the height of the cylinder is increasing four times faster than the radius. How fast is the volume of the cylinder changing? 𝑽=𝝅 𝒓 𝟐 𝒉
Vocabulary
Process and Connections
Answer and Justifications
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§2.6: Related Rates

56. Assignment
Worksheet
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§2.6: Related Rates