Related Rates
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1
What is a related rates problem?
A problem that uses one rate of change to find another for connected quantities changing over time.
2
What does t represent in related rates?
Time.
3
What does dx/dt mean?
How fast x changes over time.
4
What units does dx/dt use if x is in meters and t is in seconds?
Meters per second.
5
What does r represent in circle and sphere formulas?
Radius.
6
What does dr/dt mean?
How fast the radius changes over time.
7
What does V represent in geometry formulas?
Volume.
8
What does dV/dt mean?
How fast volume changes over time.
9
What is the difference between r and dr/dt?
r is the radius at a moment; dr/dt is its rate of change.
10
What sign does an increasing quantity’s rate have?
Positive.
11
What sign does a decreasing quantity’s rate have?
Negative.
12
What is the rate of change of a constant quantity?
Zero.
13
What are the main steps for solving a related rates problem?
Draw and label, identify rates, connect quantities with an equation, differentiate with respect to time, substitute, and solve.
14
What does differentiating with respect to time mean?
Finding how both sides of an equation change as time passes.
15
What kind of equation should you start with in related rates?
An equation connecting the quantities, such as an area formula, volume formula, or the Pythagorean theorem.
16
When should you substitute a changing quantity’s value at a particular moment?
After differentiating.
17
Why avoid substituting a changing quantity’s value before differentiating?
It makes the quantity look constant, losing its rate of change.
18
Can you substitute a truly fixed quantity before differentiating?
Yes.
19
What quantity stays fixed in a rigid sliding-ladder problem?
The ladder’s length.
20
What values and rates should you use when solving for an unknown rate?
Values and rates that apply at the same moment.
21
What is the time derivative of x² when x changes over time?
2x·(dx/dt)
22
What is the time derivative of r³ when r changes over time?
3r²·(dr/dt)
23
Why does the time derivative of x² include dx/dt?
The chain rule accounts for x changing with time.
24
What is the time derivative of xy when both variables change over time?
x·(dy/dt) + y·(dx/dt)
25
What rule differentiates xy when both variables change over time?
The product rule.
26
What is the time derivative of a fixed number such as 100?
Zero.
27
What equation connects the sides of a right triangle?
x² + y² = z², where z is the hypotenuse.
28
What is the hypotenuse of a right triangle?
The side opposite the right angle.
29
What is the time derivative of x² + y² = z²?
2x·(dx/dt) + 2y·(dy/dt) = 2z·(dz/dt).
30
What is dz/dt if the hypotenuse z stays constant?
Zero.
31
What does the triangle’s related rates equation become if z stays constant?
2x·(dx/dt) + 2y·(dy/dt) = 0.
32
What does A represent in geometry formulas?
Area.
33
What is the area formula for a circle?
A = πr².
34
What is the related rates equation for a circle’s area?
dA/dt = 2πr·(dr/dt).
35
What is the volume formula for a sphere?
V = (4/3)πr³.
36
What is the related rates equation for a sphere’s volume?
dV/dt = 4πr²·(dr/dt).
37
What is the volume formula for a cylinder?
V = πr²h.
38
What does h represent in cylinder and cone formulas?
Height.
39
How are water volume and height rates related in a cylindrical tank with fixed radius?
dV/dt = πr²·(dh/dt).
40
What is the volume formula for a cone?
V = (1/3)πr²h.
41
Why are similar triangles useful in cone and shadow problems?
They relate lengths, letting you replace one variable before differentiating.
42
What units does a length’s rate of change use?
Length per time, such as meters per second.
43
What units does an area’s rate of change use?
Square units per time, such as square meters per second.
44
What units does a volume’s rate of change use?
Cubic units per time, such as cubic meters per second.
45
If water drains at 5 cubic meters per minute, what is dV/dt?
−5 cubic meters per minute.
46
Why is dV/dt negative when water drains?
Volume is decreasing.
47
Does “the radius is 3 meters” give a rate?
No; it gives r = 3 meters, not dr/dt.
48
What factor is easy to forget when differentiating r² with respect to time?
dr/dt.
49
What is the full time derivative of r²?
2r·(dr/dt)
50
If a circle’s radius increases at 2 cm/s, what rate is given?
dr/dt = 2 cm/s.
51
What equation connects an expanding circle’s area and radius?
A = πr².
52
What equation results from differentiating A = πr² with respect to time?
dA/dt = 2πr·(dr/dt).
53
How fast does a circle’s area increase when r = 3 cm and dr/dt = 2 cm/s?
12π cm²/s.
54
In the sliding-ladder example, what does x represent?
The bottom’s distance from the wall.
55
In the sliding-ladder example, what does y represent?
The top’s height above the ground.
56
What equation connects x and y for a 10-foot ladder against a wall?
x² + y² = 100.
57
What is the time derivative of x² + y² = 100?
2x·(dx/dt) + 2y·(dy/dt) = 0.
58
How high is a 10-foot ladder’s top when its bottom is 6 feet from the wall?
8 feet.
59
What equation confirms the ladder’s height is 8 feet when x = 6 feet?
6² + 8² = 10².
60
If the ladder’s bottom moves away from the wall at 2 ft/s, what is dx/dt?
2 ft/s.
61
What rate equation applies when x = 6 feet, y = 8 feet, and dx/dt = 2 ft/s?
2·6·2 + 2·8·(dy/dt) = 0.
62
What is dy/dt when x = 6 feet, y = 8 feet, and dx/dt = 2 ft/s?
−1.5 ft/s.
63
What does dy/dt = −1.5 ft/s mean for the ladder?
The top moves downward at 1.5 feet per second.
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