Mathematics: The surface area of a sphere is decreasing at the constant rate of 3*pi sq. cm/sec . ? at what rate is the volume of the sphere decreasing at the instant its radius is 2 cm ?

The surface area of a sphere is decreasing at the - id2880950, morganluvsblueow9wtm

Ответ:

Lia1love

05.01.2022

Surface area of a sphere: $A=4\pi r^2$
Volume of a sphere: $V=\dfrac43\pi r^3$

Differentiate both with respect to an arbitrary variable for time:
$\displaystyle\frac{\mathrm dA}{\mathrm dt}=8\pi r\frac{\mathrm dr}{\mathrm dt}$
$\displaystyle\frac{\mathrm dV}{\mathrm dt}=4\pi r^2\frac{\mathrm dr}{\mathrm dt}$

You're given that $\dfrac{\mathrm dA}{\mathrm dt}=-3\pi$ when r=2

, so you can use this in the first equation to solve for $\dfrac{\mathrm dr}{\mathrm dt}$ .

$-3\pi=8\pi \times2\dfrac{\mathrm dr}{\mathrm dt}\implies\dfrac{\mathrm dr}{\mathrm dt}=-\dfrac3{16}$

Now use this to find $\dfrac{\mathrm dV}{\mathrm dt}$ .

$\displaystyle\frac{\mathrm dV}{\mathrm dt}=4\pi \times2^2\times\left(-\dfrac3{16}\right)=-3\pi$

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Ответ:

leilanimontes714

20.04.2020

Mathematics

any value for t can make this true.

Step-by-step explanation:

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It's ok if u don't...lol

I hope this helped!!

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The surface area of a sphere is decreasing at the constant rate of 3*pi sq. cm/sec . ? at what rate is the volume of the sphere decreasing at the instant its radius is 2 cm ?

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