Case 1:
If
x
≥
3, then

x

3

=
x

3 and

x
+ 2

=
x
+ 2, so the equation becomes
x
+
x
+ 2 +
x

3 = 7. Solving this gives us
x
= 8
/
3. But 8
/
3
6≥
3, so this solution
is extraneous.
Page 3
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Math 171: Final Exam
Case 2:
If

2
≤
x
≤
3, then

x

3

=

x
+ 3 but

x
+ 2

=
x
+ 2, so the equation
becomes
x
+
x
+ 2

x
+ 3 = 7. Solving this gives us
x
= 2. Since

2
≤
2
≤
3, this
solution is valid.
Case 3:
If
x
≤ 
2, then

x

3

=

x
+ 3 and

x
+ 2

=

x

2, so the equation
becomes
x

x

2

x
+ 3 = 7. Solving this gives us
x
=

6. Since

6
≤ 
2, this
solution is also valid.
So the only solutions are
x
= 2
,

6.
8. A young alien on a distant planet throws a ball into the air. It reaches its maximum
height of 30ft after 4 seconds. Assume that the ball is at 2ft when it is released at time
t
= 0.
(a) [6 points] Find a formula for the height of the ball as a function of time. (
Hint
:
The graph of your function will be a parabola.)
Solution:
We know that the formula for the height
h
at time
t
will be a
quadratic, so let
h
(
t
) =
at
2
+
bt
+
c
.
The problem gives us enough informa
tion to solve for
a
,
b
and
c
. First, we know that
h
(0) = 2, so
a
·
0
2
+
b
·
0+
c
= 2.
Therefore,
c
= 2. Second, we are told that
h
(4) = 30, so
a
·
4
2
+
b
·
4 + 2 = 30.
Simplifying, 4
a
+
b
= 7. Finally, we are told that
h
has its maximum value at
t
= 4, meaning that the parabola
y
=
h
(
t
) =
at
2
+
bt
+ 2 has its vertex there.
Completing the square,
h
(
t
) =
a
t
2
+
b
a
t
+ 2 =
a
"
t
+
b
2
a
2

b
2
a
2
#
+ 2
.
Therefore, the vertex is at
t
=

b
2
a
, so 4 =

b
2
a
. Simplifying,
b
=

8
a
. (Note
that we could have gotten this equation a little easier by either (1) using the
fact that
h
0
(4) = 0 because
h
has a local maximum at
t
= 4 or (2) that by
symmetry, after 8 seconds the ball will be at height 2ft again, so
h
(8) = 2.)
We have found that 4
a
+
b
= 7 and
b
=

8
a
.
Solving for
a
and
b
, we have
a
=

7
/
4 and
b
= 14. Thus
h
(
t
) =

7
4
t
2
+ 14
t
+ 2
.
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 Fall '07
 GOMEZ,JONES
 Math, Calculus, Algebra, Trigonometry, Sin, Quadratic equation, Mathematics in medieval Islam

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