The volume of a cube is increasing at the rate of 8cm³ / s. How fast is the surface area increasing when the length of its edge is 12 cm?

Solution:

In maths, derivatives have wide usage. They are used in many situations like finding maxima or minima of a function, finding the slope of the curve, and even inflection point

Let the side length, volume and surface area respectively be equal to x , V and S.

Hence, V = x³ and S = 6x²

We have,

dV/dt = 8cm³/ s

Therefore,

dV/dt = d/dt (x³)

8 = d/dt (x³) dx/dt

8 = 3x² dx/dt

dx/dt = 8/3x²-------(1)

Now,

dS/dt = d/dt (6x²)

= d/dx (6x²) dx/dt

= 12x dx/dt

= 12x (8/3x²) [from (1)]

So, when x = 12 cm

Then,

dS/dt = 32/12 cm²/s

= 8/3 cm²/s

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.1 Question 2

The volume of a cube is increasing at the rate of 8cm³ / s. How fast is the surface area increasing when the length of its edge is 12 cm ?

Summary:

Given that The volume of a cube is increasing at the rate of 8cm³ / s. Hence, the rate at which surface area increasing when the length of its edge is 12 cm is 8/3 cm²/s