SME
Calculus I
Related rates and optimization
Use this for “Squares are cut from the corners of a 30 cm square sheet to make an open box. What cut gives the largest volume?”
Chart
Local maximum at (5, 2,000)
Local minimum at (15, 0)
V(4) = 1,936
This is an example. Type your own values to start a new problem.
x
=
x (cm)
,
s
=
s (cm)
V
=
x(s
−
2x)²
=
V (cm³)
m
Slope of the volume V′(x) at the cut
calculated · cm²
xₘₐₓ
Best cut
worked out (not typed) · cm
Vₘₐₓ
Largest volume
worked out (not typed) · cm³
Formulas
V
=
x
×
(
s
− 2 ×
x
)
2
1,936
=
4
×
(30 − 2 × 4)
2
m
=
(
s
−
2
×
x
)
×
(
s
−
6
×
x
)
132 = (30 − 2 × 4) × (30 − 6 × 4)
xₘₐₓ
=
s
6
5
=
30
6
Vₘₐₓ
=
2
×
s
3
27
2,000
=
2
×
30
3
27
Assumptions
•
Squares of side x are cut from the four corners of an s-by-s sheet and the sides fold up: the box is x tall on a square base of side s − 2x.
•
So V(x) = x(s − 2x)², which makes sense for 0 < x < s ÷ 2. At both ends V = 0, so the largest volume is where V′ = 0 inside.
•
V′(x) = (s − 2x)(s − 6x) is 0 at x = s ÷ 2 (no box left) and at x = s ÷ 6, the largest volume.
•
Lengths are in cm and volumes in cm³; the slope V′ is in cm³ per cm of cut, which is cm².
Step-by-step
WE KNOW
s = 30 cm, x = 4 cm
FIND
volume of the box (V), slope of the volume V′(x) at the cut (m), best cut (xₘₐₓ), largest volume (Vₘₐₓ)
Step 1 · Find volume of the box (V)
Use V = x × (s − 2 × x)²
Height
x
times
the
square
base:
its
side
is
the
sheet
less
two
cuts,
s
−
2
x
.
V
=
4
×
(30 − 2 × 4)
2
V
=
1,936
cm³
Step 2 · Find slope of the volume V′(x) at the cut (m)
Use m = (s − 2 × x) × (s − 6 × x)
Multiply
V
out
and
use
the
power
rule.
V
′
>
0
here,
so
a
bigger
cut
still
adds
volume.
m
=
(30
−
2
×
4)
×
(30
−
6
×
4)
V
(
x
)
=
x
(30 − 2
x
)
2
=
4
x
³
−
120
x
²
+
900
x
V
′(
x
)
=
12
x
²
−
240
x
+
900
=
(30
−
2
x
)(30
−
6
x
)
V
′(4)
=
12
(4)
2
−
240(4)
+
900
=
132
m
=
132
cm²
Step 3 · Find best cut (xₘₐₓ)
Use xₘₐₓ = s ÷ 6
Set
V
′
=
0:
(
s
−
2
x
)(
s
−
6
x
)
=
0.
x
=
s
2
leaves
no
base,
so
the
best
cut
is
x
=
s
6
.
xₘₐₓ
=
30
6
xₘₐₓ = 5 cm
Step 4 · Find largest volume (Vₘₐₓ)
Use Vₘₐₓ = 2 × s³ ÷ 27
Put
the
best
cut
x
=
s
6
into
V
:
(
s
6
)
(
s
−
s
3
)
2
=
(
s
6
)
(
2
s
3
)
2
=
2
s
³
27
.
V
′
goes
from
+
to
−
there,
and
both
ends
give
V
=
0,
so
this
is
the
largest.
Vₘₐₓ
=
2
×
30
3
27
Vₘₐₓ = 2,000 cm³
Check
Put the numbers back in. Both sides should match.
1,936
=
4
×
(30 − 2 × 4)
2
✓
132 = (30 − 2 × 4) × (30 − 6 × 4) ✓
5
=
30
6
✓
2,000
=
2
×
30
3
27
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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