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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2020 · Mathematics · Application of Derivatives

Let l be the length and b be the breadth of a rectangle such that...

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

Options

  1. A2k22 k^2
  2. Bk2k^2
  3. Ck22\frac{k^2}{2}
  4. Dk24\frac{k^2}{4}

Correct answer

D. k24\frac{k^2}{4}

Step-by-step solution

l+b=Kl+b=K, where l=l= length, b=b= breadth. Area of the rectangle A=l⋅bA=l \cdot b. A=l(K−l)=lK−l2A=l(K-l)=l K-l^2 For maximum area, dAdl=0\frac{d A}{d l}=0ddl(lk−l2)=0K−2l=0⇒l=K2\begin{aligned} & \frac{d}{d l}\left(l k-l^2\right)=0 \\ & K-2 l=0 \Rightarrow l=\frac{K}{2} \end{aligned}$Now,Now,\frac{K}{2}+b=K \Rightarrow b=K-\frac{K}{2}=\frac{K}{2}Area,Area,A=l \cdot b=\frac{K}{2} \cdot \frac{K}{2}=\frac{K^2}{4}$
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