Moderate -0.5 This is a standard linear programming formulation question requiring students to define variables, write constraints from a table, and formulate an objective function. While it involves multiple constraints and real-world context, it follows a routine template taught in Decision Mathematics with no novel problem-solving required, making it slightly easier than average.
8 Alli is planting garlic cloves and leek seedlings in a garden.
The planting density is the number of plants that are planted per \(\mathrm { m } ^ { 2 }\)
The planting densities and costs are shown in the table below.
\(\frac{1}{16}x\) = area required for the garlic cloves
B1
Explains correctly how each term in the expression relates to the total area that Alli plants
\(\frac{1}{36}y\) = area required for the leek seedlings
Question 8(b)(i):
Answer
Marks
Guidance
Maximise \(\frac{1}{16}x + \frac{1}{36}y\)
M1
Obtains at least one correct non-trivial constraint for \(x\) or \(y\). Condone strict inequality
subject to: \(15x + 10y \leq 1500\)
A1
Obtains three correct constraints in \(x\) and/or \(y\). Condone strict inequality
\(y \geq 50\)
\(y \leq x\)
A1
Formulates the linear programming problem correctly with all constraints correct and use of 'maximise'. Condone inclusion of \(x \geq 0\)
\(x, y\) are integer
Question 8(b)(ii):
Answer
Marks
Guidance
The linear programming problem does not take into account the area of Alli's garden.
B1
Recognises a limitation of the model in the context of the problem with reference to area
Question 8(c)(i):
Answer
Marks
Guidance
\(15x + 10y \leq 1500\) has changed. This means that the number of cloves & seedlings that Alli can buy has increased as the amount of money available has increased.
E1
Evaluates the new model and identifies that the constraint modelling the money has changed
Infers a change in the financial context, such as the total money available increased or cost of cloves/seedlings has decreased, or infers implications such as Alli can now plant more cloves/seedlings
B1
Question 8(c)(ii):
Answer
Marks
Guidance
Optimal point at \((100, 50)\)
M1
Using the model, identifies a vertex of the feasible region (PI)
Calculates the correct maximum total area. AWRT 7.6 from correct working. Condone missing units
## Question 8(a):
$\frac{1}{16}x$ = area required for the garlic cloves | B1 | Explains correctly how each term in the expression relates to the total area that Alli plants
$\frac{1}{36}y$ = area required for the leek seedlings | |
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## Question 8(b)(i):
Maximise $\frac{1}{16}x + \frac{1}{36}y$ | M1 | Obtains at least one correct non-trivial constraint for $x$ or $y$. Condone strict inequality
subject to: $15x + 10y \leq 1500$ | A1 | Obtains three correct constraints in $x$ and/or $y$. Condone strict inequality
$y \geq 50$ | |
$y \leq x$ | A1 | Formulates the linear programming problem correctly with all constraints correct and use of 'maximise'. Condone inclusion of $x \geq 0$
$x, y$ are integer | |
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## Question 8(b)(ii):
The linear programming problem does not take into account the area of Alli's garden. | B1 | Recognises a limitation of the model in the context of the problem with reference to area
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## Question 8(c)(i):
$15x + 10y \leq 1500$ has changed. This means that the number of cloves & seedlings that Alli can buy has increased as the amount of money available has increased. | E1 | Evaluates the new model and identifies that the constraint modelling the money has changed
Infers a change in the financial context, such as the total money available increased or cost of cloves/seedlings has decreased, or infers implications such as Alli can now plant more cloves/seedlings | B1 |
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## Question 8(c)(ii):
Optimal point at $(100, 50)$ | M1 | Using the model, identifies a vertex of the feasible region (PI)
$\frac{1}{16} \times 100 + \frac{1}{36} \times 50 = 7.64 \text{ m}^2$ | A1 | Obtains correct coordinates of optimal vertex
| A1 | Calculates the correct maximum total area. AWRT 7.6 from correct working. Condone missing units
8 Alli is planting garlic cloves and leek seedlings in a garden.
The planting density is the number of plants that are planted per $\mathrm { m } ^ { 2 }$\\
The planting densities and costs are shown in the table below.
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2022 Q8 [10]}}