Mathematics · Oct / Nov 2023 · Paper 31 · Question 6

YMA01

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The question as text

6. An area of sea floor is being monitored. The area of the sea floor, S km 2, covered by coral reefs is modelled by the equation t S = pq where p and q are constants and t is the number of years after monitoring began. Given that log10 S = 4.5 – 0.006t

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6(a)[2]
log
S
=
4.5
−
0.006 2
×
S
=
10
4.5 0.006 2
−
×
=
30800km
2
10
6(b)[3]
log
S
=
4.5
−
0.006
t
S
=
10
4.5 0.006
−
t
(or
or
)
p =
10
4.5
q
=
10
−
0.006
10
(
)
t
S
=
10
4.5 0.006
−
t
=
10
4.5
×
10
−
0.006
(or
and
)
p =
10
4.5
q
=
10
−
0.006
S =
31600 (0.986)t
×
6(c)[1]
E.g. The proportion of area covered by coral reefs retained from year to year.
Notes
(a)
M1
Substitutes in
t =
2
and proceeds to find a value for S via
10 May be implied by a correct
...
value for their equation. Note awrt 24000 may arise from a slip with 0.006×2 = 0.12, and is
acceptable for M1.
A1:
awrt
30800km including units, from correct work and isw as long as units are included with a
2
correct value. But e.g. 30760 followed by 307 km2 is A0.
(b)
or award for either (
)
or (
)
M1:
Writes
S
=
10
4.5 0.006
−
t
p =
10
4.5
q
=
10
−
0.006
even if from an incorrect
step. Essentially for knowing they need to raise to base 10. May arise from attempts at solving
simultaneous equations using the answer to (a) (allowing for rounding errors).
(
)
t
dM1: Writes
S
=
10
4.5
×
10
−
0.006
or award for both
p =
10
4.5
and
q
=
10
−
0.006
(oe forms e.g accept
1
1
or
for q) Must be fully correct method to this point but allow if all that is
10
0,006
1.01
wrong is rounding errors from other methods such solving simultaneous equations (e.g. 0.990t).
A1cso:
S =
31600 (0.986)t
×
Must be awrt 3 s.f. values. Must be an equation including the S, not just p
and q values, but accept if seen in (c).
(c)
B1:
E.g. accept the following or variations of them
The proportion of area covered by coral reefs retained from year to year.
The seafloor covered by coral reduces by 1.4% each year.
The proportional change/decrease in sea floor area covered by coral reefs per year
Do not accept “rate of change” or “decrease in area of reefs per year” or “q is the gradient” as
the answer as it is an incorrect interpretation. The proportional change should be conveyed.
There must be reference to, or equivalents of, the following:
sea floor area or sea floor or area or seabed
coral reefs or coral or reef,
each year or year to year or per year
(q or its value is) a percentage/proportional decrease/remaining
Some acceptable answers:
After 1 year the area will reduce to S last year times q. It shows the speed that coral reefs seafloor reduce.
The percentage of remaining area of sea floor covered by coral reefs to the area of that covered last year.
q represents the proportion of decreasing of the area of sea floor covered by coral reefs every year.
Some unacceptable answers:
When there is extra t, the S will increase one more unit of 0.986
q means the percentage of the part of sea floor covered by coral reefs after t years.
q is the amount of coral reefs die/shrink in accordance to time.
The area of the sea floor is decreasing at almost 0.986 times before.
For every increase in the value of t, the total area of coral reefs will decrease in a rate of 0.986t

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