The volume of a cube is increasing at the rate of 9 cm3/s. Consider t…

The correct answer is Option (4) → (A), (B) and (D) only

Let the edge be a cm. Differentiating \(V = a^3\) with respect to time (chain rule):

    \( \displaystyle \frac{dV}{dt} = 3a^2\frac{da}{dt} \)

    \( \displaystyle \frac{da}{dt} = \frac{9}{3a^2} = \frac{3}{a^2} \)

The surface area is \(S = 6a^2\) and the total edge length is L = 12a:

    \( \displaystyle \frac{dS}{dt} = 12a\frac{da}{dt}, \quad \frac{dL}{dt} = 12\frac{da}{dt} \)

At a = 10: \(\frac{da}{dt} = \frac{3}{100} = 0.03\), \(\frac{dS}{dt} = 120 \times 0.03 = 3.6\), \(\frac{dL}{dt} = 12 \times 0.03 = 0.36\).

At a = 20: \(\frac{da}{dt} = \frac{3}{400} = 0.0075\).

So at a = 10 the edge grows at 0.03 cm/s, the surface area at 3.6 cm²/s and the total edge length at 0.36 cm/s; at a = 20 the edge grows at 0.0075 cm/s.

Checking the statements

(A) When the edge is 10 cm, the edge is increasing at 0.03 cm/s. — Correct. \(\frac{da}{dt} = \frac{9}{3 \times 100} = 0.03\).

(B) When the edge is 10 cm, the surface area is increasing at 3.6 cm²/s. — Correct. \(\frac{dS}{dt} = 12 \times 10 \times 0.03 = 3.6\).

(C) When the edge is 10 cm, the total length of all the edges is increasing at 0.03 cm/s. — Incorrect. All 12 edges grow together: \(12 \times 0.03 = 0.36\) cm/s.

(D) When the edge is 20 cm, the edge is increasing at 0.0075 cm/s. — Correct. \(\frac{da}{dt} = \frac{9}{3 \times 400} = 0.0075\).