The volume of a cube is increasing at the rat | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $x$ be the length of an edge,$V$ be the volume,and $S$ be the surface area of the cube.We have $V = x^{3}$ and $S = 6x^{2}$.Given that the rat

Vedclass · Accepted Answer

$\frac{8}{3} \, cm^{2}/s$

Vedclass · Answer

(A) Let $x$ be the length of an edge,$V$ be the volume,and $S$ be the surface area of the cube.We have $V = x^{3}$ and $S = 6x^{2}$.Given that the rate of change of volume is $\frac{dV}{dt} = 8 \, cm^{3}/s$.Using the chain rule,$\frac{dV}{dt} = \frac{d}{dt}(x^{3}) = 3x^{2} \cdot \frac{dx}{dt}$.Substituting the given values: $8 = 3x^{2} \cdot \frac{dx}{dt} \implies \frac{dx}{dt} = \frac{8}{3x^{2}}$.Now,the rate of change of surface area is $\frac{dS}{dt} = \frac{d}{dt}(6x^{2}) = 12x \cdot \frac{dx}{dt}$.Substituting $\frac{dx}{dt}$ from the previous step: $\frac{dS}{dt} = 12x \cdot \left(\frac{8}{3x^{2}}\right) = \frac{32}{x}$.When the edge length $x = 12 \, cm$,the rate of change of surface area is $\frac{dS}{dt} = \frac{32}{12} = \frac{8}{3} \, cm^{2}/s$.

The volume of a cube is increasing at the rate of $8 \, cm^{3}/s$. How fast is the surface area increasing when the length of an edge is $12 \, cm$?

Similar Questions

The side of a square sheet of metal is increasing at the rate of $3 \text{ cm/min}$. At what rate is the area increasing when the length of the side is $6 \text{ cm}$?

The distance travelled $s$ (in $cm$) by a particle in $t$ seconds is given by $s = t^3 + 2t^2 + t$. The speed of the particle after $1$ second will be ......... $cm/sec$.

$A$ circular plate of radius $5 \text{ cm}$ is heated. Due to expansion,its radius increases at the rate of $0.05 \text{ cm/s}$. The rate at which its area is increasing when the radius is $5.2 \text{ cm}$ is:

Find the rate of change of $\sqrt{x^2 + 16}$ with respect to $\frac{x}{x - 1}$ at $x = 3$.

Two cyclists are moving from a junction of two roads inclined at an angle of $120^{\circ}$ to each other,with velocities of $4 \text{ km/h}$ and $3 \text{ km/h}$ respectively. The rate at which they are separating from each other after $1 \text{ hour}$ is ..... $\text{km/h}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module