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The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 40 mm

Sagot :

Answer:

[tex]\dfrac{dV}{dt}=502.65\ mm^3/s[/tex]

Step-by-step explanation:

The volume of a sphere is given by :

[tex]V=\dfrac{4}{3}\pi r^3[/tex]

The rate of change of volume means,

[tex]\dfrac{dV}{dt}=\dfrac{d}{dt}(\dfrac{4}{3}\pi r^3)\\\\\dfrac{dV}{dt}=\dfrac{4}{3}\pi \times 3r\times \dfrac{dr}{dt}[/tex]

We have, [tex]\dfrac{dr}{dt}=2\ mm/s\ and\ r=40\ mm[/tex]

So,

[tex]\dfrac{dV}{dt}=\dfrac{4}{3}\pi \times 3\times 20\times 2\\\\=502.65\ mm^3/s[/tex]

So, the volume is increasing at the rate of [tex]502.65\ mm^3/s[/tex].

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