An architect designs a garden in a circular shape with centre E and radius 50 m. There is a flower bed ABCD of rectangular shape inside the garden as shown in the figure. Suppose length and width of the flower bed are 2x and 2y meters respectively.
Based on above information answer the following question:
The maximum area (in m2) of the flower bed is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5000
Half the diagonal of the flower bed is the radius, so by Pythagoras’ theorem \(x^2 + y^2 = 2500\), and
\(A(x) = 4xy = 4x\sqrt{2500 - x^2}\)
Differentiating:
\( \displaystyle A'(x) = 4\sqrt{2500-x^2} - \frac{4x^2}{\sqrt{2500-x^2}} \)
\( \displaystyle = \frac{8(1250 - x^2)}{\sqrt{2500-x^2}} \)
A′(x) = 0 gives
\(x^2 = 1250,\quad x = 25\sqrt{2}\)
A′(x) changes sign from positive to negative there, so this is a maximum. Then \(y = \sqrt{2500 - 1250} = 25\sqrt{2}\) as well:
\(A_{\max} = 4(25\sqrt{2})(25\sqrt{2}) = 4 \times 1250 = 5000\)