Physics

08.05.2022 08:54

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The volume of a sphere is increasing at a rate of 25\pi25π25, pi cubic meters per

The volume of a sphere is increasing at a rate of 25\pi25π25, pi cubic meters per hour. at a certain instant, the volume is \dfrac{32\pi}{3} 3 32π start fraction, 32, pi, divided by, 3, end fraction cubic meters.

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simran922

4,6(3 marks)

$\dfrac{dA}{dt}=25\pi$

Explanation:

given,

rate of increase of volume of sphere = 25π m³/h

volume at certain time = $\dfrac{32\pi}{3}$

calculate the rate of change of surface area = ?

volume of sphere

$V = \dfrac{4}{3}\pi r^3$

$\dfrac{32\pi}{3} = \dfrac{4}{3}\pi r^3$

r³ = 8

r = 2 m

surface are of the sphere

A = 4 π r²

$\dfrac{dA}{dr}=\dfrac{d}{dr}(4\pi r^2)$

$\dfrac{dA}{dr}=8\pi r$

$\dfrac{dV}{dr}=\dfrac{d}{dr}(\dfrac{4}{3}\pi r^3)$

$\dfrac{dA}{dr}=4\pi r^2$

now,

$\dfrac{dA}{dt}=\dfrac{dA}{dr}\dfrac{dr}{dV}\dfrac{dV}{dt}$

$\dfrac{dA}{dt}=8\pi r \dfrac{1}{4\pi r^2}\dfrac{dV}{dt}$

$\dfrac{dA}{dt}=\dfrac{2}{r}\dfrac{dV}{dt}$

$\dfrac{dA}{dt}=\dfrac{2}{2}\times (25\pi)$

$\dfrac{dA}{dt}=25\pi$

calebbailey2318

5,0(38 marks)

D. Mechanical energy remains unchanged.

Mechanical energy = Potential energy + Kinetic energy

So if potential energy decreases, then kinetic energy must increase so that the system is balanced.

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