Mathematics · May / June 2021 · Paper 21 · Question 9
YMA01
As printed in the paper
The question as text
9. h cm l cm l cm Figure 3 Figure 3 shows a sketch of a square based, open top box. The height of the box is h cm, and the base edges each have length l cm. Given that the volume of the box is 250 000 cm3
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
9(a)[3]
V
=
hl ⇒
2
25000
0
=
hl
2
or
l
=
oe (may use e.g.
hl =
250000h
)
h
"
250000
"
"
500
"
S
l
2
4
hl
4
h
=
+
=
+
×
h
h
250000
S
=
+
2000 h
*
h
9(b)[5]
d
S
250000
1
1
−
2000
h
oe
=
−
+
×
2
d
2
2
h
h
d
S
250000
1
1
−=
=
0
⇒
−
+
2000
×
h
2
0
⇒
h
k
=
..
.
d
h
h
2
2
3
⇒
h
2
=
250
⇒
h
=
..
.
2
h =
250
3
3
9(c)[2]
d
2
S
500000
−
=
−
500
h
2
d
h
2
h
3
d
2
S
=
6
>
0
hence gives the minimum value.
d
h
2
h
=
39.7
Notes:
(a)
B1: Correct equation linking h and l seen or implied by substitution.
M1: Attempts the surface area and substitutes to eliminate l. Allow
l
2
+
2(
hl
+
hl
)
or
2(
l
2
(or equivalents) for the attempt at surface area.
A1*: Achieves the given result with no errors.
(b)
Allow if d
S
h is missing or called d
y
x throughout.
d
d
M1: Attempts derivative of the area function – power decreased by one in at least one term.
A1: Correct derivative. Need not be simplified.
dM1: Sets derivative equal to zero and attempts to solve as far as a power of h equal to something.
Depends on first M
ddM1: Depends on both M marks, attempts to solve an equation with a fractional index to find h
2
2
A1:
h =
250
or states
k =
3
3
(c)
3
A
−
M1: Attempts the second derivative, form
−
Bh
reached (A,B > 0). Note: the question specifies
2
h
3
by further differentiation, so first derivative test attempts score M0.
A1: Second derivative correct and substitutes h to get awrt 6, and makes correct conclusion referring
to sign (>0 or positive). Must see the substitution as it is not obviously positive but allow if
expression is not fully simplified as long as it is clear it is positive (e.g. 8 – 2 > 0).
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