A creative landscaper wants to lay concrete kerb stones around the perimeter for his flowerbeds as s

Question

A creative landscaper wants to lay concrete kerb stones around the perimeter for his flowerbeds as shown in the diagram below.

The flowerbed is made up of a rectangle $PQRS$ with a semicircle on the outer side of $PS$ and a quadrant of a circle on the outer side of $QR$.

The length of $PS$ is $2x$ m and the length of $PQ$ is $y$ m. The total perimeter of the flowerbed is $50$ m.

Show that the area, $A$ $\mathrm{m}^2$, of the flowerbed is given by \[A = 50x - \frac{1}{2}(4 + \pi)x^2.\][3]

The landscaper would like to maximise the area of the flowerbed.

TGM adapted task: use a non-calculator method to find the maximum of the quadratic expression in (a) over real $x$. Determine whether that value is attainable by a flowerbed with $x>0$ and $y>0$. Explain what happens to the area as $y$ tends to zero from above.[3]

TGM source note: the original part (b) asks for a maximum flowerbed area without checking positive lengths. The adapted task above deliberately asks you to test that modelling assumption; the source dimensions and part (a) are unchanged.

Accepted Answer

(a) $A=50x-\frac12(4+\pi)x^2$. (b) The quadratic has maximum $\frac{1250}{4+\pi}$, but its maximiser gives $y<0$. For the stated non-degenerate flowerbed there is no attained maximum; the supremum is $\frac{1875\pi}{2(1+\pi)^2}\text{ m}^2$. See the geometry check in the worked solution.

2024 VJC H1 Q2

2024 VJC H1 Q2

Solution:

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