Part II: The Answers
324
Answers
501–600
You can verify that this value gives you a maximum by using the first derivative test.
Using 5x + 2y = 1,500 and x = 150 gives you y = 375; therefore, the area is
A = ( )(
) =
150 375 56 250
,
ft
2
502.
16 ft
3
If a square with side x is removed from each corner of the piece of cardboard, you have
the following volume, where 0 < x < 3:
V
x
x x
= ( − )( − )
6 2 6 2
By distributing, you have
V
x
x x
x
x x
x
x
x
= ( − )( − )
=
−
+
=
−
+
6 2 6 2
36 24
4
4
24
36
2
3
2
(
)
You want the maximum volume, so take the derivative of this function:
′
V
x
x
=
−
+
12
48
36
2
Then set the derivative equal to zero and solve for x:
12
48
36 0
12
4
3 0
12
3
1 0
3 1
2
2
x
x
x
x
x
x
x
−
+
=
−
+ =
−
− =
=
(
)
(
)(
)
,
Because x = 3 makes the volume zero, you know that x = 1 gives you a maximum,
which you can verify by using the first derivative test.
Using V = (6 – 2x)(6 – 2x)x and x = 1 gives you the volume:
V ( )
1
4
16
= ( )(4)(1)
=
ft
3