OCR H240/02 2019 June — Question 4 5 marks

Exam BoardOCR
ModuleH240/02 (Pure Mathematics and Statistics)
Year2019
SessionJune
Marks5
PaperDownload PDF ↗
TopicExponential Functions
TypeCompare or choose between models
DifficultyModerate -0.5 This question requires interpreting graphs and equations of exponential models, but involves no calculation—only qualitative description of behavior. The tasks are straightforward: describing rate of growth from graphs, analyzing long-term behavior from given equations (recognizing exponential decay/growth patterns), and making contextual interpretations. While it requires understanding of exponential functions, it's primarily reading comprehension and pattern recognition rather than technical manipulation, making it slightly easier than average.
Spec1.02z Models in context: use functions in modelling1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context
4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}
  1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
    The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
  2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
    1. using model A,
    2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
  3. State what is suggested about the long-term food supply by
    1. model A,
    2. model B.
Question 4:
Part (a)
AnswerMarks Guidance
Answer/WorkingMark Guidance
A: Growth (rate) increases, then decreases. Grows slowly, then quickly, then slowly. B: Growth (rate) decreases. Grows quickly then slowly. Both.B1 Allow increase, constant, then decrease or "levels off", "tails off", "plateaus"; Allow "levels off", "tails off", "plateaus"; BOD if describe growth rather than rate in (a) and (b); NOT "\(P\)" decreases, for A or B; Ignore "exponentially"; Condone muddle between \(P\) and growth of \(P\) in (a) and (b)
Part (b)(i)
AnswerMarks Guidance
Answer/WorkingMark Guidance
A: \(P\) (decreases and) tends to 20, or (Decreases and) doesn't go below 20B1 Allow (Decrease and) reach 20; Must mention 20 (as population, not years); Ignore all else
Part (b)(ii)
AnswerMarks Guidance
Answer/WorkingMark Guidance
B: \(P\) tends to 1020 oe; \(P\) doesn't exceed 1020B1 Growth is asymptotic around 1020; Settles at 1020; Saturates at 1020; Converges to 1020; Allow reaches 1020; Plateaus at 1020; Asymptote at 1020; Must mention 1020; NOT: Pop increases, but slowly; Diverges to 1020; Tends to 1020, then down; Ignore all else
Part (c)(i)
AnswerMarks Guidance
Answer/WorkingMark Guidance
A: Food (almost) runs out, or is used up oe, or becomes very low or there will be a shortage oe, or begins to run outB1 or will only support a population of 20; Won't sustain large nos.; Insufficient; NB "Limited" allowed in c(ii), not c(i); NOT: just Limited, Finite; NOT: just "Decreases"; Ignore all else
Part (c)(ii)
AnswerMarks Guidance
Answer/WorkingMark Guidance
B: Food sufficient to support a pop \(\approx 1020\); Enough to sustain equilibrium (or population); Barely enough, can't support increase in \(P\); Food limited so pop can't continue to growB1 Stays stable; Sustainable; Constant; Must imply at least two of: 1. Food won't run out and 2. Food limited or equilibrium 3. Can't support increase in \(P\); Ignore all else 
# Question 4:

## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| A: Growth (rate) increases, then decreases. Grows slowly, then quickly, then slowly. B: Growth (rate) decreases. Grows quickly then slowly. Both. | B1 | Allow increase, constant, then decrease or "levels off", "tails off", "plateaus"; Allow "levels off", "tails off", "plateaus"; **BOD if describe growth rather than rate in (a) and (b)**; NOT "$P$" decreases, for A or B; Ignore "exponentially"; Condone muddle between $P$ and growth of $P$ in (a) and (b) |

## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| A: $P$ (decreases and) tends to 20, or (Decreases and) doesn't go below 20 | B1 | Allow (Decrease and) reach 20; **Must mention 20** (as population, not years); Ignore all else |

## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| B: $P$ tends to 1020 oe; $P$ doesn't exceed 1020 | B1 | Growth is asymptotic around 1020; Settles at 1020; Saturates at 1020; Converges to 1020; Allow reaches 1020; Plateaus at 1020; Asymptote at 1020; **Must mention 1020**; NOT: Pop increases, but slowly; Diverges to 1020; Tends to 1020, then down; Ignore all else |

## Part (c)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| A: Food (almost) runs out, or is used up oe, or becomes very low or there will be a shortage oe, or begins to run out | B1 | or will only support a population of 20; Won't sustain large nos.; Insufficient; NB "Limited" allowed in c(ii), not c(i); NOT: just Limited, Finite; NOT: just "Decreases"; Ignore all else |

## Part (c)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| B: Food sufficient to support a pop $\approx 1020$; Enough to sustain equilibrium (or population); Barely enough, can't support increase in $P$; Food limited so pop can't continue to grow | B1 | Stays stable; Sustainable; Constant; Must imply at least two of: 1. Food won't run out and 2. Food limited or equilibrium 3. Can't support increase in $P$; Ignore all else |

---
4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population $P$ after $t$ years. The diagrams show these models as they apply to the first 20 years.\\
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242}\\
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}
\begin{enumerate}[label=(\alph*)]
\item Without calculation, describe briefly how the rate of growth of $P$ will vary for the first 20 years, according to each of these two models.

The equation of the curve for model A is $P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }$.\\
The equation of the curve for model B is $P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)$.
\item Describe the behaviour of $P$ that is predicted for $t > 20$
\begin{enumerate}[label=(\roman*)]
\item using model A,
\item using model B .

There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
\end{enumerate}\item State what is suggested about the long-term food supply by
\begin{enumerate}[label=(\roman*)]
\item model A,
\item model B.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR H240/02 2019 Q4 [5]}}
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