Mathematics · Oct / Nov 2021 · Paper 31 · Question 7
YMA01
As printed in the paper
The question as text
7 The mass, M kg, of a species of tree can be modelled by the equation log10M = 1.93 log10r + 0.684 where r cm is the base radius of the tree. The base radius of a particular tree of this species is 45 cm. According to the model,
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7(a)[2]
log
M
=
1.93log 45
+
0.684
⇒
log
M
=
3.8747
10
10
10
(M =) awrt 7500 (kg)
7(b)[3]
log
M
=
1.93log
r
+
0.684
q
M
=
p r
10
10
1.93
q
log
M
log
r
0.684
log
M
log
pr
=
+
=
10
10
10
10
1.93
q
log
r
+
0.684
log
M
=
log
p
+
log
r
M
10
=
10
10
10
10
log
log
log
0.684
1.93
M
=
p
+
q
r
M
=
10
×
r
10
10
10
0.684
Either
p =
10
or
q =
1.93
p =
awrt 4.83
q =
1.93
Both
and
7(c)[1]
''p'' is the mass (in kg) of a tree with radius 1 cm
(a)
M1
Substitutes r = 45 in
log
M
=
1.93log
r
+
0.684
and proceeds to
log
M =
10
10
10
(which must be a numerical value) (Condone the use of r = 0.45).
Implied by awrt 7500 for r = 45 or awrt 1.0 for r = 0.45
A1
(M =) awrt 7500 (kg). Condone
7.5 10
×
3
Alt (a)
q
M1
Substitutes r = 45 in their
M
=
p r
and finds M.
A1
(M =) awrt 7500 (kg)
(b) Mark (b) and (c) together
q
B1
A correct proof showing that
log
M
=
1.93log
r
+
0.684
⇔
M
=
p r
. Condone
10
10
log or lg for
log but use of natural logs ln is B0
10
Most are starting with
log
M
=
1.93log
r
+
0.684
. Expect to see the addition (or
10
10
subtraction) law of logs explicitly used or index law being applied before proceeding to
the final answer.
(
)
Eg
log
M
log
4.83
log
r
1.93
log
4.83
r
1.93
M
4.83
r
1.93
is B1
=
+
=
×
⇒
=
10
10
10
10
log
M
=
log
r
1.93
+
0.684
⇒
M
=
10
1.93log
r
+
0.684
⇒
M
=
10
1.93log
r
×
10
0.684
10
10
10
10
log
M
=
log
r
1.93
+
0.684
⇒
M
=
10
1.93log
r
+
0.684
⇒
M
=
r
1.93
×
10
0.684
=
4.83
10
10
10
log
M
=
log
r
1.93
+
0.684
⇒
M
=
10
1.93log
r
×
10
0.684
=
4.83
r
1.93
is B1
10
10
10
log
M
=
log
4.83
+
log
r
1.93
⇒
M
=
4.83
r
1.93
is B0 (addition law of logs not seen)
10
10
10
1.93
1.93
log
r
+
0.684
log
r
0.684
No incorrect working should be seen such as
10
10
=
10
10
+
10
but condone the omission of the brackets around
log
4.83× r
1.93
if seen in their working.
10
0.684
q
M1
Either
p =
10
or
q =
1.93
This may be implied by
M
=
p r
with a ''correct'' p
or q
A1
Both
p =
awrt 4.83
and
q =
1.93
(B0M1A1 can be scored) This may be implied
1.93
by
M
=
4.83
r
(c)
B1
''p'' is the mass of a tree with radius 1 cm
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