[related rates][section 12]

(1.)  xy = 6 and dx/dt = 5            given

Find dy/dt when x = 3.      here is the problem

x(dy/dt) + y(dx/dt) = 0   take the derivative implicitly

3y = 6            replace x with 3
__  ___
3    3          divide each side by 3

y = 2           divide and cancel

3(dy/dt) + 2(5) = 0    make  substitutions

3(dy/dt) + 10 = 0       multiply

-10  -10     subtract 10 from each  side
_______________________
3(dy/dt)     = -10      subtract
_________      ____
3             3         divide each side by 3

dy/dt = -10/3                  cancel

(2.)   Given x/y = 2 and dx/dt = 4;   Find dy/dt when

x = 2.

2/y = 2             replace x with 2

2 = 2y             multiply each side by y, cancel
___  ____
2     2              divide each side by 2

1 = y               divide and cancel

y(dx/dt) - x(dy/dt)
______________________ = 0    take the derivative
y2                      implicitly

1(4) - 2(dy/dt)
____________________ = 0       make substitutions
12

4 - 2(dy/dt) = 0           multiply and divide

-4 + 2(dy/dt) = 0        multiply thru by -1

+    4            +  4       add 4 to each side
_______________________
2(dy/dt) = 4          add
________  ___
2       2    divide each side by 2

dy/dt = 2      divide and cancel

(3.)  A 10-foot ladder is leaning against the side of a

house.   As the foot of the ladder is pulled away from

the house, the top of the ladder slides down along the

side of the house.   Suppose the foot of the ladder is

pulled away at a rate of 2 ft/sec.   How fast is the top

of the ladder sliding down when the foot is 8 ft from

the house?

[use a 6 - 8 - 10 pythagorean triangle]

a2 + b2 = c2     use the pythagorean theorem

2a(da/dt) + 2b(db/dt) = 2c(dc/dt)   take the derivative

implicitly

a(da/dt) + b(db/dt) = c(dc/dt)  divide thru by 2, cancel

6(da/dt) + 8(2) = 10(0)    make substitutions

6(da/dt) + 16 = 0             multiply

-     16  -16     subtract 16 from each side
_____________________________
6(da/dt)  =  -16            subtract
________    _____
6        6         divide each side by 6

da/dt = -8/3         reduce and cancel

(4.)  Two roads intersect at right angles.  A car travelling

80 km/hr reaches the intersection half an hour before a

bus that is travelling on the other road at 60 km/hr.  How

fast is the distance between the car and the bus increasing

1 hour after the bus reaches the intersection?

a2 + b2 = c2      use the pythagorean theorem

1202 + 602 = c2    replace a with 120 and b with 60

14,400 + 3600 = c2           square

18,000 = c2             add like terms

(3600)(5) = c2       factor like this
_
c = 60
5    take the square root of each side

2a(da/dt) + 2b(db/dt) = 2c(dc/dt)

[take the derivative implicitly]

a(da/dt) + b(db/dt) = c(dc/dt)  divide thru by 2, cancel
_
(120)(80) + (60)(60) = (60
5)(dc/dt)  make substitutions
_
7200 + 3600 = (60
5)(dc/dt)      multiply
_
10,800 = (60
5)(dc/dt)    add like terms
________  _____________
_          _
60
5       605          divide each side by this
_
180/
5 = dc/dt         reduce and cancel
_                                        _
dc/dt = 180
5/5          multiply top and bottom by 5
_
dc/dt = 36
5                divide

(5.)  A sailor standing on the edge of a dock 15 ft above the

lake surface is pulling in his boat by means of a line

attached to the boat's bow.  He pulls in the line at

a rate of 5 ft/min.  How fast is the boat approaching the

foot of the dock when the boat is 20 ft away?

Use a 15 - 20 - 25   right triangle.

a2 + b2 = c2    use the pythagorean theorem

2a(da/dt) + 2b(db/dt) = 2c(dc/dt)    take the derivative

implicitly

a(da/dt) + b(db/dt) = c(dc/dt)   divide thru by 2, cancel

15(0) + 20(db/dt) = 25(5)    make substitutions

20(db/dt) = 125             multiply add
_________  _____
20       20            divide each side by 20

db/dt = 6.25     divide and cancel

(6.)  An airplane at a height of 1000 m is flying horizontally

at a velocity of 500 km/hr and passes directly over an

observer.   How fast is the plane receding from the

observer when it is 1500 m away, (along a direct line

of sight), from the observer?

a2 + b2 = c2     use the pythagorean theorem

10002 + b2 = 15002    replace a with 1000 & c with 1500

1,000,000 + b2 = 2,250,000    square 1000 and 1500

-1,000,000      -1,000,000  subtract this fr ea side
__________________________________
b2 = 1,250,000       subtract

b2 = (5)(250,000)    factor like this
_
b = 500
5    take square roots

a2 + b2 = c2    use the pythagorean theorem again

2a(da/dt) + 2b(db/dt) = 2c(dc/dt)   take the derivative

implicitly

a(da/dt) + b(db/dt) = c(dc/dt)   divide thru by 2

1000(0) + (500
5)(500,000) = 1500(dc/dt)

[make substitutions]
_
(500
5)(500,000) = 1500(dc/dt)   multiply, add
_
(5
5)(5000) = 15(dc/dt)   divide thru by 100
_
25,000
5 = 15(dc/dt)     multiply
__________  _________
15        15          divide each side by 15
_
(5000/3)
5 = dc/dt       reduce and cancel

(7.)  Sand is dropped onto a conical pile at a rate of 15 m3/min.

Suppose the height of the pile is always equal to its

diameter.  How fast is the height increasing when the

pile is 7 m high?

V = (1/3)(pi)r2h          use this formula

V = (1/3)(pi)(h/2)2h    replace r with h/2

V = (h3/12)(pi)            multiply

dV/dt = (h2/4)(pi)(dh/dt)    take the derivative implicitly

15 = (72/4)(pi)(dh/dt)   replace dV/dt with 15 & h with 7

15 = (49/4)(pi)(dh/dt)      square 7

60 = 49(pi)(dh/dt)        multiply each side by 4, cancel
___  ______________
49pi    49pi               divide each side by 49pi

60
_________ = dh/dt               cancel
49pi

(8.)  A rock is thrown into a pool of water.  A circular wave

leaves the point of impact and travels so that its radius

increases at a rate of 25 cm/sec.  How fast is the

circumference of the wave increasing when the radius is

1 m?

C = 2
r    use the circumference formula

dC/dt = 2
∏(dr/dt)    take the derivative implicitly

dC/dt = 2∏(25)
make substitutions

dC/dt = 50
multiply

(9.)  Bacteria grow in circular colonies.  The radius of one

colony is increasing at the rate of 4 mm/day.  On

Wednesday, the radius of the colony is 1 mm.  How fast

is the area of the colony changing one week later?

A = ∏r2     use the area formula for circle

dA/dt = 2∏r(dr/dt)   take the derivative implicitly

dA/dt = 2∏(7)(4)    replace r with 7, replace dr/dt with 4

dA/dt = 56∏                multiply

(10.)  A spherical mothball is dissolving at a rate of 8pi

cm3/hr.  How fast is the radius of the mothball de-

creasing when the radius is 3 cm.

V = (4/3)
∏r3          use the volume formula

dV/dt = 4
∏r2(dr/dt)   take the derivative implicitly

8∏ = 4∏(3)2(dr/dt)   replace dV/dt with 8∏

replace r with 3

8∏ = 36∏(dr/dt)         multiply
____  __________
36∏    36∏             divide each side by 36∏

2/9 = dr/dt           reduce, cancel