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A#39
Name:___________________________________ Date:________ Period:____
Review#5 Worksheet: Implicit and Related Rates
1. As shown in the figure below , water is draining from a conical tank with height 12 feet and
diameter 8 feet into a cylindrical tank that has a base with area 400  square feet. The depth h,
in feet of the water in the conical tank is changing at the rate of ( h - 12 ) feet per minute.
( The volume of a cone with radius r and height h is V =
1 2
r h )
3
a) Write an expression for the volume of water in the conical tank as a function of h.
b) At what rate is the volume of water in the conical tank changing when h = 3? Indicate units
of measure
c) Let y be the depth, in feet , of the water in the cylindrical tank. At what rate is y changing
when h = 3? Indicate units of measure
8
12
h
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2. The volume of an expanding sphere is increasing at a rate of 12 cubic feet per second.
When the volume of the sphere is 36 cubic feet, how fast in square feet per second is the
surface area increasing?
4r 3
Note: ( V =
and S = 4r2 )
3
3. If x2 - 3xy + 10 = 0, find the value of “y” when the tangent to the curve is vertical and
when the tangent line is horizontal.
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4. Find the derivative of cos ( x + y ) = x
5. Find the derivative of x2 + 9xy + 4x – 3xy – 7 = 0
6. Find the equation of the tangent and normal line of
x2 – xy = -20 when y = 9 .
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7. Write the equation of the tangent and normal line for the curve x2 = cot (xy) at the
 
point  ,1 .
2 
8. Find the slope of the curve xy = 5y2 + 7y where y = 1.
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9. Find the coordinates of the points where vertical and horizontal tangents exist for
4xy – 3y2 = 2 .
10. Water is being drained from an inverted conical tank at a rate of 12 cubic feet per
second. At the instant the height is 12, the volume of the cone is 124. At this instant
how fast is the radius decreasing.
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11. Given x3 – xy + y2 = 3
dy y  3x 2
a) Show

dx 2 y  x
b) Write the equation of the tangent line where x = 1.
c) Find the x-coordinates of any horizontal tangents.
0
ANSWERS:

3
dv
 9 ft
min
dt
2
ds
 8 ft
sec
27
dt
dy 1  sin( x  y )
2 10

3. No vertical tangent; horizontal tangents when y  
4.
dx
 sin( x  y )
3
dy  x  3 y  2
1
 x  4 and y  9  1 x  5

5.
6. Tangents: y  9 
dx
3x
4
5

  2  2 
Normal: y  9  4x  4 and y  9  5x  5 7. Tangent: y  1  
 x  

2


1

  
Normal: y  1  
8.
9. Vertical:  3 , 2    3 , 2  ;
 x  
3 
2
3
 2
5
2
 2  2 
No horizontal tangent
10. The radius is decreasing at a rate of 0.180 ft/sec.
4
1
x  1 c) x = -0.710, 0.822
11. b) y  1   x  1 and y  2 
3
3
1. a) V 
h3
b)
c)
9 ft
min
400
2.