Let l be the length and b be the breadth of a rectangle such that l + b = k. What is the minimum area of rectangle?

Question

Let $l$ be the length and $b$ be the breadth of a rectangle such that $l + b = k .$ What is the minimum area of rectangle?

Smart Achievers · Accepted Answer

(4)

Ans. $\frac{k^{2}}{4}$

Explanation:

Here, $l + b = k$

or $l = k - b$

Area, $A = l \times b = (k - b) b$

$A = k b - b^{2}$

$∴$ $\frac{d A}{d b} = k - 2 b$

For maximum area,

$\frac{d A}{d b} = 0$

$\Rightarrow$ $k - 2 b = 0$

$\Rightarrow$ $b = \frac{k}{2}$

$\frac{d^{2} A}{d b^{2}} = - 2 < 0$ (maximum area)

Now,

$A = (k - b) \times b = (k - \frac{k}{2}) \times \frac{k}{2} = \frac{k^{2}}{4}$

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