MATH241 Lecture 21: Related Rates
9 May 2018
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MATH241 - Lecture 21 - Related Rates
3.9: Related Rates
If two functions are related by an equation, then their derivatives are also going to be related.
The general approach is to compute the rate of change of one quantity in terms of another.
Example
:
If A is the area of a circle with radius r,
A) Find in terms of
dt
dA
dt
dr
B) If the radius increases at a constant rate of 2 m/s, how fast is the area increasing when
m?0r= 1
Solution:
A) rA = π 2
dt
dA =d
dt (πr)
2
→ (Chain Rule)πr
dt
dA = 2 dt
dr
B) ,
dt
dr = 2 0r= 1
→ π 0
dt
dA = 2 · 1 · 2
0π= 4 /sm2
Example
:
Air is pumped into a spherical balloon at a rate of . How fast is the radius of the00 m/s12
balloon increasing when ? mr = 2
Solution:
πrV =3
43
dt
dV =d
dt
πr
[3
4 3]
πr= 4 2
dt
dr
00
dt
dV = 1 r= 2
00 π1 = 4 · 22
dt
dr
→ 00 6π 1 = 1 dt
dV
→ dt
dV =16π
100
m/s=25
4π
Example
:
A 15 foot ladder rests against a vertical wall. If the bottom of the ladder slides away from the
wall at a rate of 2 ft/s, how fast is the top of the ladder moving when the bottom is 9 ft from
the wall?
Solution:
x2+y2= 152
x y 2 dt
dx + 2 dt
dy = 0
,
dt
dx = 2 x= 9
x2+y2= 152
92+y2= 152
→ 1 258 + y2= 2
→ 44y2= 1
→ 2y= 1
x y 2 dt
dx + 2 dt
dy = 0
2 (9) (2) + 2 (12) dt
dy = 0
→ 6 4 3 + 2 dt
dy = 0
→ 4 62 dt
dy = − 3
→ dt
dy = − 24
36
ft/s= − 2
3
Document Summary
If two functions are related by an equation, then their derivatives are also going to be related. The general approach is to compute the rate of change of one quantity in terms of another. If a is the area of a circle with radius r: find da dt in terms of dr dt. If the radius increases at a constant rate of 2 m/s, how fast is the area increasing when r = 1 m?0. A = 2 r da = d dt ( r )2 dt dr. 0 da = 2 1 2. 0 da = 2 dt dr = 2 dt. Air is pumped into a spherical balloon at a rate of balloon increasing when r = 2 m. A 15 foot ladder rests against a vertical wall. Solution: x2 + y2 = 152 dx + 2 dt x y. 2 dt dx = 2 x = 9 dt dy = 0.
