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[25 POINTS] A cylinder has a volume of 400π cm3. Find the dimensions that minimize its surface area.

[25 POINTS] A cylinder has a volume of 400π cm3. Find the dimensions that minimize-example-1

1 Answer

12 votes

Answer:

r = 5.848cm, h = 11.696cm

Explanation:

Surface area:
SA=2\pi rh+2\pi r^2

Volume:
V=\pi r^2h

Therefore, the height of a cylinder given its volume would be
h=(V)/(\pi r^2), thus:


SA=2\pi r((V)/(\pi r^2))+2\pi r^2\\\\SA=(2V)/(r)+2\pi r^2\\\\SA=(2(400\pi))/(r)+2\pi r^2\\\\SA=(800\pi)/(r)+2\pi r^2

We now find the derivative of the surface area of the cylinder with respect to its radius and set it equal to 0, solving for the radius:


(d(SA))/(dr)=-(800\pi)/(r^2)+4\pi r


0=-(800)/(r^2)+4\pi r


0=800\pi+4\pi r^3


-800\pi=4\pi r^3


-200=r^3


r\approx5.848

If
0If [tex]r>5.848, then the surface area of the cylinder increases

Therefore, the surface area is minimized when the radius is
r=5.848cm, making the minimum height
h=(V)/(\pi r^2)=(400\pi)/(\pi(5.848)^2)\approx11.696cm.

In conclusion, the 2nd option is correct.

User Christian Achilli
by
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