Part II: The Answers
332
Answers
501–600
A x
x
x
x
x
x
( )
(
)
=
+
−
−
=
+
−
4
1
2
20
3
3
2
20
3
16
3
36
20
2
2
2
( ) ( ) ( )
where 0 ≤ x ≤ 20.
Find the derivative of this function:
′
A x
x
x
x
x
( )
(
)( )
(
)
=
+
−
−
=
−
−
2
16
3
36
2 20
1
1
8
3
18
20
Set the derivative equal to zero and solve for x:
1
8
3
18
20
0
1
8
10 3
9
3
18
0
9
72
80 3
72
4 3
72
0
9
72
4 3
72
x
x
x
x
x
x
x
x
−
−
=
−
+
=
−
+
=
+
(
)
= =
+
=
= +
80 3
72
9 4 3
72
80 3
72
80 3
9 4 3
x
x
It’s reasonable to let x = 0 (the wire is used entirely to make the triangle) or x = 20 (the
wire is used entirely to make the square), so check those values as well. Here are the
areas:
A
A
0
0
16
3
36
20 0
3
36
400 19 25
20
20
16
3
36
20 20
2
2
2
2
( ) = +
−
(
) = ( )≈
( )=
+
−
(
)
.
= =
=
+
=
+
+
−
+
400
16
25
80 3
9 4 3
80 3
9 4 3
16
3
36
20
80 3
9 4 3
2
A
≈
2
10 87
.
Therefore, the minimum occurs when
x = +
≈
80 3
9 4 3
8 70
. meters
511.
20 5
1 5
3
3
+
feet from the bright light source
332
Answers
501–600
A x
x
x
x
x
x
( )
(
)
=
+
−
−
=
+
−
4
1
2
20
3
3
2
20
3
16
3
36
20
2
2
2
( ) ( ) ( )
where 0 ≤ x ≤ 20.
Find the derivative of this function:
′
A x
x
x
x
x
( )
(
)( )
(
)
=
+
−
−
=
−
−
2
16
3
36
2 20
1
1
8
3
18
20
Set the derivative equal to zero and solve for x:
1
8
3
18
20
0
1
8
10 3
9
3
18
0
9
72
80 3
72
4 3
72
0
9
72
4 3
72
x
x
x
x
x
x
x
x
−
−
=
−
+
=
−
+
=
+
(
)
= =
+
=
= +
80 3
72
9 4 3
72
80 3
72
80 3
9 4 3
x
x
It’s reasonable to let x = 0 (the wire is used entirely to make the triangle) or x = 20 (the
wire is used entirely to make the square), so check those values as well. Here are the
areas:
A
A
0
0
16
3
36
20 0
3
36
400 19 25
20
20
16
3
36
20 20
2
2
2
2
( ) = +
−
(
) = ( )≈
( )=
+
−
(
)
.
= =
=
+
=
+
+
−
+
400
16
25
80 3
9 4 3
80 3
9 4 3
16
3
36
20
80 3
9 4 3
2
A
≈
2
10 87
.
Therefore, the minimum occurs when
x = +
≈
80 3
9 4 3
8 70
. meters
511.
20 5
1 5
3
3
+
feet from the bright light source