Question

the volume of the spherical ball is increasing at the rate of 4πcc/sec.
find the rate of the radius and the surface area are changing when the volume is 288π cc.​

Answer 1

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The volume of the spherical ball is increasing at the rate of 4πcc/sec. Find the rate of the radius and the surface area are changing when the volume is 288π cc.

$\huge\bf\boxed{\boxed{\underline{\red{Answes!!}}}}$

Volume of a spherical ball increase at the rate of = 4πcc/s

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Now, we are required to find the rate of the increase of the surface area when the volume is

= 288π

$\huge\mathrm\</em><em>p</em><em>i</em><em>n</em><em>k</em><em>{</em><em>\</em><em>f</em><em>r</em><em>a</em><em>c</em><em>{</em><em>dA</em><em>}</em><em>{</em><em>dt</em><em>}</em><em> </em><em>=</em><em> </em><em>?</em><em>¿</em><em>}$

Now :

Since the volume of the sphere is = 288 π

we can find the radius = !??

using the volume of a sphere is $\frac{</em><em>4</em><em>}{</em><em>3</em><em>}</em><em>πr³</em><em>$

➪ $\frac{4}{3}πr³$ = 288π

taking Reciprocal :

➪ r³ = 288 × $\frac{3}{4}$

➪ r³ = 216

➪ r = 6

Now :

To find $\frac{</em><em>dv</em><em>}{dt}$

☞︎︎︎ $\frac{dv}{dt}$ = $\frac{dv}{dr}$ × $\frac{dr}{dt}$

Differentiate the volume of the sphere gives us:

➪ $\frac{dv}{dt}$ = $\frac{4}{3}πr³$ ×3r²

denominator and numerator 3 get cancel :

➪ 4πr²

Now :

to find $\frac{dr}{dt}$ :

☞︎︎︎ $\frac{dv}{dt}$ = $\frac{dv}{dr}$ × $\frac{dr}{dt}$

➪ 4πr² = $\frac{4}{3}πr³$ × $\frac{dr}{dt}$

➪ $\cancel\frac{4π}{4πr²}$ × = $\frac{dr}{dt}$

➪ $\frac{dr}{dt}$ = $\frac{1}{r²}$

Now the next process :

to fine $\frac{</em><em>dA</em><em>}{dt}$

☞︎︎︎ $\frac{dA}{dt}$ = $\frac{dA}{dr}$ × $\frac{dr}{dt}$

Remember, the surface area of the sphere is

Remember, the surface area of the sphere is 4πr²

So :

☞︎︎︎ A = 4πr²

☞︎︎︎ $\frac{dA}{dt}$ = 8πr

We already found that r = 6 and $\frac{dr}{dt}$ = $\frac{1}{r²}$

Putting this all together,

☞︎︎︎ $\frac{dA}{dt}$ = $\frac{dA}{dr}$ × $\frac{dr}{dt}$

➪ $\frac{dA}{dt}$ = 8πr × $\frac{1}{r²}$

➪ $\frac{dA}{dt}$ = $\frac{8πr}{r²}$

r and r get cancel :

➪ $\frac{dA}{dt}$ = $\frac{8π}{r}$

➪ $\frac{dA}{dt}$ = $\frac{8π}{6}$

➪ $\frac{dA}{dt}$ = $\frac{4}{3}π$

$\huge\bf\boxed{\boxed{\underline{\red{</em><em>Ans</em><em> </em><em>=</em><em> </em><em>\</em><em>f</em><em>r</em><em>a</em><em>c</em><em>{</em><em>4</em><em>}</em><em>{</em><em>3</em><em>}</em><em>π</em><em>}}}}$

Answer 2

gudd bye .... thank you for being all that time with me....now it's my time to go...as there is a hii there is always a bye....so I have to go now...time spend with you all was really amazing and I enjoyed a lot here...thank you for everything... BORAHAEE ♥️

I will never forget u ಥ_ಥ

GOODBYE...

-aizah