Mathematics · Oct / Nov 2022 · Paper 21 · Question 4
YMA01
As printed in the paper
The question as text
4. The weight of a baby mammal is monitored over a 16‑month period. The weight of the mammal, w kg, is given by 2 t 18 w = loga(t + 5) – loga4 where t is the age of the mammal in months and a is a constant. Given that the weight of the mammal was 10 kg when t = 3
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[3]
10
=
log 8 log 4
−
log 2
=
10
a
a
a
a
10
=
2
a =
2
110
=
1.07177...
*
4(b)[3]
t
+
5
w
=
log
(
t
+
5)
−
log
4
w
=
log
1.072
1.072
1.072
4
t
+
5
t
+
5
w
log
1.072
w
t
...
=
=
=
1.072
4
4
t = ×
4 1.072
w
−
5
(b)
w
=
log
(
t
+
5)
−
log
4
w
+
log
4
=
log
(
t
+
5)
ALT
1.072
1.072
1.072
1.072
+
t
5
=
1.072w
+
log
4
1.072
1.072
w
+
log
4
−
5
=
t
1.072
4(c)[2]
t = ×
4 1.072
15
−=
5
...
awrt 6.35
(a)
8
M1: Substitutes
and
into the equation and achieves log 2
=
10
or e.g.
log
=
10
t =
3
w =
10
a
a
4
correctly
M1: Correctly removes the log to obtain
a
10
=
"2"
10 8
10 2
A1*: Fully correct proof showing
a =
2
110
or
a =
or
a =
and obtains awrt 1.072
4
Note that this may be implied by the accuracy of their a e.g. 1.071773463… so allow for
1.0718 (rounded) or 1.0717 (truncated).
log2
log2
a
10
2
log
a
10
log2
log
a
a
10
May also see logarithm approach e.g.
=
=
=
=
10
10
ln 2
ln2
10
10
or
a
=
2
ln
a
=
ln 2
ln
a
=
a
=
e
10
=
...
10
False solutions in (a):
log 8
10
=
log 8
−
log 4
a
=
10
log 2
=
10
a
10
=
2
a
=
10
2
=
1.072
a
a
log 4
a
a
Scores M0M1A0
8log
a
10
=
log 8
−
log 4
=
10
2log
a
=
10
log
a
2
=
10
a
10
=
2
a
=
10
2
=
1.072
a
a
4log
a
Scores no marks
(b)
t
+
5
M1: Applies the subtraction law for logs to write the equation as
w
=
log
1.072
4
( )
M1: Writes the equation as
1.072
w
=
f
t
and proceeds to make t the subject.
A1:
t = ×
4 1.072
w
−
5
Alternative:
M1: Rearranges to make
log
(
t +
5)
the subject correctly
1.072
M1: Removes the logs on lhs to obtain ( )
( )
f
t =
1.072
g
w
A1: Correct equation.
1.072
w
+
log
4
5
t
=
−
1.072
In both cases allow the use of “a” or a more accurate value for “a” rather than 1.072 for both
method marks.
(c)
M1: Substitutes
w =
15
into their equation from (b) or possibly the given equation
w
=
log
(
t
+
5)
−
log
4
and proceeds to find a value for t
1.072
1.072
A1: awrt 6.35 (months) (NB If full accuracy used for a answer is 6.3137… and scores A0)
Ignore any units if given.
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