Amongest all open(from the top)right circular cylinderical boxes of vol.125 pie cm cube.Find the dimensions of the box which has the least surface area.
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The dimensions of the box which has the least surface are radius = 5 cm and height = 5 cm .
Given that the volume of the right circular cylindrical box = 125π cm³.
Let the radius of the cylinder be r and the height be equal to h.
=> Volume = πr²h
=> πr²h = 125π
=> r²h = 125
=> h = 125/r²
Surface area of the box = 2πrh + πr²
= 2π r × (125/r²) + πr²
= (250π/r) + πr²
Let f(r) = (250π/r) + πr² be a function w.r.t r.
Differentiating f(r) w.r.t r to find the point of minima .
f'(r) = (-250π/r²) + 2πr = 0
=> 2πr = 250π/r²
=> r³ = 125
=> r = 5 cm
h = 125/25 = 5 cm
The dimension of the cylindrical box is radius = 5 cm and height = 5 cm .
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Answer:
5cm is your answer.........
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