A balloon,which always remains spherical,has | vedclass

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

Question

(B) The volume of a sphere $(V)$ with radius $(r)$ is given by $V = \frac{4}{3} \pi r^3$.Given the diameter $d = \frac{3}{2}(2x+1)$,the radius $r$ is

Vedclass · Accepted Answer

$\frac{27}{8} \pi(2x+1)^2$

Vedclass · Answer

(B) The volume of a sphere $(V)$ with radius $(r)$ is given by $V = \frac{4}{3} \pi r^3$.Given the diameter $d = \frac{3}{2}(2x+1)$,the radius $r$ is $r = \frac{d}{2} = \frac{3}{4}(2x+1)$.Substituting $r$ into the volume formula:$V = \frac{4}{3} \pi \left( \frac{3}{4}(2x+1) \right)^3 = \frac{4}{3} \pi \cdot \frac{27}{64} (2x+1)^3 = \frac{9}{16} \pi (2x+1)^3$.To find the rate of change of volume with respect to $x$,we differentiate $V$ with respect to $x$:$\frac{dV}{dx} = \frac{d}{dx} \left( \frac{9}{16} \pi (2x+1)^3 \right) = \frac{9}{16} \pi \cdot 3(2x+1)^2 \cdot \frac{d}{dx}(2x+1)$.$\frac{dV}{dx} = \frac{27}{16} \pi (2x+1)^2 \cdot 2 = \frac{27}{8} \pi (2x+1)^2$.

$A$ balloon,which always remains spherical,has a variable diameter $\frac{3}{2}(2x+1)$. Find the rate of change of its volume with respect to $x$.

Similar Questions

$A$ ladder of length $13 \ m$ has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \ m/min$, then the speed (in $m/min$) at which the upper end falls when the bottom is $5 \ m$ away from the wall is

The total revenue received from the sale of $x$ units of a product is given by $R(x) = 10x^2 + 20x + 1500$. When $x = 1500$,the marginal revenue is . . . . . . .

The displacement of a particle in time $t$ is given by $s = 2t^2 - 3t + 1$. The acceleration is

$A$ man of $5 \text{ feet}$ height is walking away from a light fixed at a height of $15 \text{ feet}$ at the rate of $K \text{ miles/hour}$. If the rate of increase of his shadow is $\frac{11}{5} \text{ feet/sec}$, then $K=$ (Take $1 \text{ mile} = 5280 \text{ feet}$)

$A$ ladder makes an angle of $30^\circ$ with the wall. $A$ man is climbing the ladder at the rate of $3 \text{ ft/sec}$. What is the rate at which he is approaching the wall?

Vedclass Test Series

Exam Paper Generator

Online Exam Module