# Given that the volume of a cylinder is 250 cm^3, and the surface area must be less than or equal to 400 cm^2, how do you find the radius and height in order for the surface area to be minimal without using derivatives?

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Given that the volume of a cylinder is 250 cm^3, and the surface area must be less than or equal to 400 cm^2, how do you find the radius and height in order for the surface area to be minimal without using derivatives?

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pi r^2 h= 250pi

So h= 250/r^2

A= 2pi rh +2 pir^2

A= 2pi r 250/r^2 +2pi r^2= 500 pi/r+2pi r^2

dA/dr= -500pi/r^2 +4pi r

Set equal to 0 and solve for r

500pi/r^2= 4pi r

r^3=125; r=5

h= 250/r^2= 10.
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Let radius = r cm and height = h cm

Volume = πr^2 h = 250π, h = 250/r^2

Area A = 2πrh + 2πr^2 = 2π(rh+r^2)

= 2π(250/r + r^2) = 2π(250r^-1+ r^2)

dA/dr = 2π(-250r^-2 + 2r) =0 for maxima /minima

2r = 250/r^2

2 r^3 = 250

r^3 = 125

r = 5 cm

h = 250/25 = 10 cm
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Let r and h be the radius and height f the cylinder respectively.

Volume, V= πr²h

Given volume = 250 π

250 π= πr²h

r²h=250

h= 250/r²

Surface area, A = 2πr²+2πrh

A= 2πr²+ 2πr(250/r²)

A= 2πr²+ (500π/r)

dA/dr = 4πr+500π(-1/r²)

For A to be maximum or minimum, dA/dr = 0

4πr-500(π/r²) = 0

r = 5

h = 250/(5)² = 10

By second derivative test,

d²A/dr² = 4π-500π(-2/r³)

At r= 5, d²A/dr² = 4π+1000(π/r³) > 0 . It is a local minimum.

Ans: Radius = 5 and Height = 10