Calculus

Application of Derivatives

Medium

core

The edge of a cube is increasing at a rate of 7cm/s. The rate of change of area of the cube when edge of the cube is 3cm is:

42 cm²/s

168 cm²/s

84 cm²/s

252 cm²/s

✅ Correct Option: 4

The edge of a cube is increasing at a rate of $7$ cm/s, so $\frac{da}{dt} = 7$ cm/s.

The edge length at this moment is $a = 3$ cm.

A cube has 6 faces, each face is a square with area $a^2$ .

Surface Area $A = 6a^2$

Differentiating both sides with respect to time $t$ :

$\frac{dA}{dt} = 6 \times 2a \times \frac{da}{dt}$

$\frac{dA}{dt} = 12a \times \frac{da}{dt}$

Substituting $a = 3$ cm and $\frac{da}{dt} = 7$ cm/s:

$\frac{dA}{dt} = 12 \times 3 \times 7$

$\frac{dA}{dt} = 36 \times 7$

$\frac{dA}{dt} = 252$ cm²/s

Therefore, the rate of change of surface area of the cube is $252$ cm²/s.

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