Math 10B Final Exam Solutions
150 pts
December 8, 2009
Professor Evans
Directions:
Answers alone are not sufficient.
Justify, and show all work.
The multiple part
problems 4,6,7 are worth 18 points each. The other six problems are worth 16 points each.
(1)
A man was murdered in a 34.6 degree Fahrenheit meat locker. At noon, the corpse temper
ature was 38.6, and an hour later, it had dropped to 36.6. The man was murdered at
(a) 6 am
(b) 7 am
(c) 8 am
(d) 9 am
(e) none of these times
Solution
: Let
T
(
t
) denote the temperature of the corpse at time
t
, with noon being chosen
as the “zero hour”
t
= 0.
The solution to the differential equation
dT/dt
=
k
(34
.
6

T
) is
T
(
t
) = 34
.
6 +
Be

kt
. Plug in
t
= 0 to get 38
.
6 = 34
.
6 +
B
, so
B
= 4 and
T
= 34
.
6 + 4
e

kt
. Plug
in
t
= 1 to get 36
.
6 = 34
.
6 + 4
e

k
. Thus
e

k
= 1
/
2 and
T
= 34
.
6 + 4(1
/
2)
t
. At the time of the
murder, 98
.
6 = 34
.
6 + 4(1
/
2)
t
, so 16 = (1
/
2)
t
. Thus
t
=

4, so the murder took place at 8 am.
(2)
An upright vase of height 16 is formed by rotating the curve
y
=
x
4
(0
≤
x
≤
2) around
the
y
axis. Using an integral, find the volume of water that it takes to fill the vase.
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 Summer '10
 reed
 irvine station

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