a closed right circular cylinder has volume 2156 cubic units what could be the radius of the base so that the total surface area maximum

Let r and h be the radius and height of the right circular cylinder respectively.
Now, we know that the volume of the cylinder isπr2h. So,
πr2h=2156     h=2156πr2                           ...(1)

Next, let the total surface area of the cylinder be S. So,
S=2πr2+2πrh  =2πr2+2πr2156πr2                     (Using (1))  =2πr2+4312r

Differentiating the total surface area with respect to r, we get
dSdr=4πr-4312r2
Also,
d2Sdr2=4π+8624r3
 
Now,
dSdr=04πr-4312r2=04πr3-4312r2=0
4πr3=4312r3=43124πr=4312×74×223
r=3433r=7

And
d2Sdr2r=7=4π+8624343>0
Therefore, the total surface area of the cylinder is maximum when radius is 7 units.

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