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

$A$ spherical iron ball of radius $10 \ cm$ is coated with a layer of ice of uniform thickness. The ice melts at a rate of $50 \ cm^3/min$. When the thickness of the ice is $5 \ cm$,the rate at which the thickness of the ice decreases is:

The area of a square increases at the rate of $0.5 \text{ cm}^2/\text{sec}$. The rate at which its perimeter is increasing when the side of the square is $10 \text{ cm}$ long,is

If $r$ is the radius of a spherical balloon at time $t$ and the surface area of the balloon changes at a constant rate $K,$ then ....

$A$ ladder $5 \ m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground,away from the wall,at the rate of $2 \ cm/s$. How fast is its height on the wall decreasing when the foot of the ladder is $4 \ m$ away from the wall?

Find the rate of change of the area of a circle with respect to its radius $r$ when $r = 3 \text{ cm}$. (in $\pi \text{ cm}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module