325
Answers
501–600
Answers and Explanations
503.
20 4
3
cm × 20 4
3
cm × 10 4
3
cm
If you let x be a length of one side of the base and let y be the height of the box,
the volume is
V x x y
x y
=
=
( )( )
,
16 000
2
The square base has an area of x
2
, and the box has four sides, each with an area
of xy, so the surface area is given by
S x y
x
xy
( , ) =
+
2
4
You can write this as a function of one variable by using x
2
y = 16,000 to get
y
x
=
16 000
2
,
. Substituting the value of y into the surface area formula gives you
S x
x
x
x
x
x
( )
,
,
=
+
=
+
−
2
2
2
1
4
16 000
64 000




You want the minimum surface area, so take the derivative of this function:
′
S
x
x
x
x
=
−
=
−
−
2
64 000
2
64 000
2
3
2
,
,
Then set the derivative equal to zero and solve for x:
2
64 000 0
2
64 000
32 000
8 000 4
20 4
3
3
3
3
3
x
x
x
x
x
−
=
=
=
=
=
,
,
,
( ,
)( )