The edge of a cube is increasing at the rate of 2 cm/s. Match List-I with List-II, where List-II gives each rate at the instant when the edge is 3 cm (in cm, cm2 or cm3 per second).
| List-I (Quantity) | List-II (Rate of change) | ||
| (A) | Volume of the cube | (I) | 24 |
| (B) | Total surface area of the cube | (II) | \(2\sqrt{2}\) |
| (C) | Length of a diagonal of one face | (III) | 54 |
| (D) | Total length of all the edges | (IV) | 72 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
Let x be the edge at time t, with \(\frac{dx}{dt} = 2\) and x = 3. Each quantity is differentiated with respect to t by the chain rule.
(A) Volume of the cube
This matches (III) 54.
\( \displaystyle V = x^{3},\ \frac{dV}{dt} = 3x^{2}\frac{dx}{dt} = 3 \times 9 \times 2 = 54 \)
(B) Total surface area of the cube
This matches (IV) 72.
\( \displaystyle S = 6x^{2},\ \frac{dS}{dt} = 12x\frac{dx}{dt} = 12 \times 3 \times 2 = 72 \)
(C) Length of a diagonal of one face
This matches (II) \(2\sqrt{2}\).
\( \displaystyle l = \sqrt{2}\,x,\ \frac{dl}{dt} = \sqrt{2}\frac{dx}{dt} = 2\sqrt{2} \)
(D) Total length of all the edges
This matches (I) 24.
\( \displaystyle p = 12x,\ \frac{dp}{dt} = 12\frac{dx}{dt} = 24 \)