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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 4 · Set 1 · Q38
Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as…
Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in the figure. He has $\displaystyle 200$ metres of fencing wire.
Based on the above information, answer the following questions :
(i)
Let ' $\displaystyle \mathrm{x}$ ' metres denote the length of the side of the garden perpendicular to the brick wall and ' $\displaystyle y$ ' metres denote the length of the side parallel to the brick wall. Determine the relation representing the total length of fencing wire and also write $\displaystyle \mathrm{A}(\mathrm{x})$, the area of the garden.
(ii)
Determine the maximum value of $\displaystyle \mathrm{A}(\mathrm{x})$.
Marking-scheme solution
(i)(a)
$\displaystyle 2x + y = 200$
(b)
$\displaystyle A(x) = xy = x(200 - 2x)$
(ii)
From (a) and (b) of (i) we have
$\displaystyle A(x) = x(200 - 2x)$
$\displaystyle = 200x - 2x^2$
From max./min of $\displaystyle A(x)$
$\displaystyle \dfrac{dA}{dx} = 0$ i.e., $\displaystyle 200 - 4x = 0$
$\displaystyle \Rightarrow x = 50.$
$\displaystyle \dfrac{d^2A}{dx^2} = -4$
$\displaystyle \left(\dfrac{d^2A}{dx^2}\right)_{x=50} < 0$
Hence, $\displaystyle A(x)$ is maximum at $\displaystyle x = 50$
Thus, Max $\displaystyle A(x) = 200(50) - 2(50)^2$
$\displaystyle = 10000 - 5000$
$\displaystyle = 5000$ sqm.
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