Question

Volume of a Cube The volume V of a cube with sides of length x in. is changing with respect to time. At a certain instant of time, the sides of the cube are 7 in. long and increasing at the rate of 0.2 in./s. How fast is the volume of the cube changing (in cu in/s) at that instant of time?

Answer 1

Answer:

Therefore the volume of cube is change at the 29.4 cube in./s at that instant time.

Explanation:

Formula

• $\frac{dx^n}{dx} =nx^{n-1}$

Cube :

The volume of a cube is = $side^3$

The side of length is x in.

Then volume of the cube is (V) = $x^3$

∴ V = $x^3$

Differentiate with respect to t

$\frac{d}{dt}(V)=\frac{d}{dt} (x^3)$

$\Rightarrow \frac{dV}{dt} =3x^2\frac{dx}{dt}$....(1)

Given that the side of the cube is increasing at the rate of 0.2 in/s.

i.e $\frac{dx}{dt} = 0.2$  in/s.

And the sides of the cube are 7 in i.e x= 7 in

Putting $\frac{dx}{dt} = 0.2$  and  x= 7 in equation (1)

$\therefore \frac{dV}{dt} =3 \times 7^2 \times 0.2$  cube in./s

       =29.4 cube in./s

Therefore the volume of cube is change at the 29.4 cube in./s at that instant time.