If an open box with a square base is to be made out of a given card board of area \( K^2 \) square units, then the maximum volume of the box is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{K^3}{6\sqrt{3}} \) cubic units
Let the base side be x and the height h. An open box has a base and four sides, so
\( \displaystyle x^2+4xh=K^2 \)
\( \displaystyle h=\frac{K^2-x^2}{4x} \)
\( \displaystyle V=x^2h=\frac{K^2x-x^3}{4} \)
\( \displaystyle \frac{dV}{dx}=\frac{K^2-3x^2}{4}=0 \)
\( \displaystyle x=\frac{K}{\sqrt{3}} \)
Since \( \frac{d^2V}{dx^2}=-\frac{3x}{2}\lt 0 \), V is maximum there.
\( \displaystyle V_{\max}=\frac{1}{4}\left(\frac{K^3}{\sqrt{3}}-\frac{K^3}{3\sqrt{3}}\right) \)
\( \displaystyle =\frac{1}{4}\cdot\frac{2K^3}{3\sqrt{3}} \)
\( \displaystyle =\frac{K^3}{6\sqrt{3}}\ \text{cubic units} \)
NCERT Exemplar:
Application of Derivatives
