A box is to be made with a square base and an | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the side of the square base be $x$ and the height of the box be $y$.Given that the surface area of the box with an open top is $S = x^2 + 4xy

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(C) Let the side of the square base be $x$ and the height of the box be $y$.Given that the surface area of the box with an open top is $S = x^2 + 4xy = 48$.From this,we can express $y$ in terms of $x$: $4xy = 48 - x^2 \Rightarrow y = \frac{48 - x^2}{4x}$.The volume of the box is $V = x^2y$.Substituting the value of $y$,we get $V = x^2 \left( \frac{48 - x^2}{4x} ight) = \frac{1}{4}(48x - x^3)$.To find the maximum volume,we differentiate $V$ with respect to $x$: $\frac{dV}{dx} = \frac{1}{4}(48 - 3x^2)$.Setting $\frac{dV}{dx} = 0$,we get $48 - 3x^2 = 0 \Rightarrow x^2 = 16 \Rightarrow x = 4$ (since $x > 0$).Now,find $y$ using $x = 4$: $y = \frac{48 - 4^2}{4(4)} = \frac{48 - 16}{16} = \frac{32}{16} = 2$.The maximum volume is $V = x^2y = 4^2 	imes 2 = 16 	imes 2 = 32 \, m^3$.

$A$ box is to be made with a square base and an open top. If the area of the material used is $48 \, m^2$,then the maximum volume of the box is ........... $m^3$.

Similar Questions

The number of points of local maxima and local minima of the function $f(x) = |x^2 - 2|x||$ in $\mathbb{R}$ are $M$ and $m$ respectively. Then,the value of $2M + m$ is -

The maximum value of the function $f(x) = 3x^3 - 18x^2 + 27x - 40$ on the set $S = \{x \in R : x^2 + 30 \leq 11x\}$ is

$A$ sector is removed from a metallic disc and the remaining region is bent into the shape of a circular conical funnel with volume $2 \sqrt{3} \pi$. The least possible diameter of the disc is

The set of all real values of $\lambda$ for which the function $f(x) = (1 - \cos^2 x)(\lambda + \sin x)$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ has exactly one maxima and exactly one minima is

At what value of $x$ does the function $f(x) = \sin x(1 + \cos x)$ have a maximum value?

Vedclass Test Series

Exam Paper Generator

Online Exam Module