AP Calculus AB Student Sample Question 5
AP? CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES
Question 5
The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height h,
the radius of the funnel is given b y r
1 20
3 h2
,
where
0h
10.
The units of
r
and
h
are inches.
(a) Find the average value of the radius of the funnel.
(b) Find the volume of the funnel.
(c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is
h
3
inches, the
radius of the surface
of the liquid is
decreasing at a rate of
1 5
inch per second. At this
instant, what is the rate of change of the height of the liquid with respect to time?
(a)
Average
radius
1 10
10 0
1 20
3 h2
dh
1 200
3h
h3 10 3 0
1 200
30
1000 3
-0
16009 in
3
:
1 1
: :
integral antiderivative
1 : answer
(b)
Volume 010
1 20
3 h2
2
dh
400
10 9 6h2 h4
0
dh
400
9h
2h3
h5 5
10 0
400
90
2000
100000 5
-0
2209 40
in3
3
:
1 1
: :
integrand antiderivative
1 : answer
(c)
dr dt
1 20
2h
dh dt
-
1 5
130
dh dt
dh dt
- 1 10 53
- 23 in/sec
{ 2 : chain rule
3: 1 : answer
? 2016 The College Board. Visit the College Board on the Web: .
? 2016 The College Board. Visit the College Board on the Web: .
? 2016 The College Board. Visit the College Board on the Web: .
? 2016 The College Board. Visit the College Board on the Web: .
? 2016 The College Board. Visit the College Board on the Web: .
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