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 Option 1: √3K^3/6 cubic units Option 2: K^3/3√6 cubic units Option 3: √6K^3/3 cubic units Option 4: K^3/6√3 cubic units

Answer: K^3/6√3 cubic units. Solution: Let the base side be x and height h. An open box uses x^2+4xh=K^2, so h=(K^2-x^2)/4x and V=x^2h=(K^2x-x^3)/4. Then V'=(K^2-3x^2)/4=0 gives x=K/√3, and substituting back gives V=K^3/6√3.

Calculus

Application of Derivatives

Hard

core

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

$\dfrac{\sqrt{3}K^3}{6}$ cubic units

$\dfrac{K^3}{3\sqrt{6}}$ cubic units

$\dfrac{\sqrt{6}K^3}{3}$ cubic units

$\dfrac{K^3}{6\sqrt{3}}$ cubic units

✅ Correct Option: 4

Let the base side be $x$ and height $h$ . An open box uses $x^2+4xh=K^2$ , so $h=\frac{K^2-x^2}{4x}$ and $V=x^2h=\frac{K^2x-x^3}{4}$ . Then $V'=\frac{K^2-3x^2}{4}=0$ gives $x=\frac{K}{\sqrt{3}}$ , and substituting back gives $V=\frac{K^3}{6\sqrt{3}}$ .

2025: 22 May Shift 1

The function $f(x) = x^2e^{-2x}$ increases on

2026: 15 May Shift 1

Match the LIST-I with LIST-II

	LIST-I (Function)		LIST-II (Critical point)
A.	$y = x^2 + x + 1$	I.	$2/3$
B.	$y = 2x^3 - 6x^2 + 6x$	II.	$-1$
C.	$y = 2x^2 + 4x - 1$	III.	$-1/2$
D.	$y = x^3 - x^2$	IV.	$1$

Choose the correct answer from the options given below:

2025: 14 May Shift 2

The radius of spherical balloon is decreasing at the rate of 0.1cm/sec, the rate at which its volume is decreasing, when its radius is 0.5cm is

2026: 25 May Shift 1

The maximum value of $f(x) = x^{50} - x^{20}$ in the interval $[0,1]$ , is

2024: 16 May Shift 1

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$ , $x \in [0, 3]$ , match List-I with List-II :

List-I	List-II
A. Absolute maximum value	(I) $3$
B. Absolute minimum value	(II) $0$
C. Point of maxima	(III) $-5$
D. Point of minima	(IV) $4$

2026: 22 May Shift 2

The cost function of a product is given by $C(x)=\frac{1}{3}x^3-45x^2-900x+36$ , where x is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced?

If an open box with a square base is to be made out of a given card board of area K2K^2K2 square units, then the maximum volume of the box is

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CUET Mathematics Past Year Questions (Topic-Wise)

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