Answer to Question #7929 in Differential Equations for DAN

Question #7929
A company wants to manufacture an open cylindrical bucket of volume 10 litres (10000 cm3). The plastic used for the base of the bucket costs 0.03 cents per cm2 while the plastic used for the curved side of the bucket costs 0.02 cents per cm2. Find the radius and height of the bucket for which the bucket has minimum cost. What is the minimum cost? Show all the reasoning and evaluate your answers to 2 decimal places.
Expert's answer
Let's denote radius as r and height as h. Then the area of the base is pi*r^2. The area of the side is 2*pi*r*h.
The total cost of material used for the bucket is pi*r^2*0.03+2*pi*r*h*0.02.

Volume of the bucket is V=pi*r^2*h=10000 => h=10000/pi/r^2
f(r)=pi*r^2*0.03+2*pi*r*10000/pi/r^2 *0.02=pi*r^2*0.03+2*10000/r *0.02
Now we need to find extremum if this let's find its derivative:
f'(r)=pi*r*0.06-2*10000/r^2 *0.02=0
=> pi*r*0.06-2*10000/r^2 *0.02=0 whence r=12.85
That's minimum point(because f''(12.85)>0).
So we have radius r=12.85 cm,
heught h=10000/pi/r^2=10000/pi/12.85^2=19.29 cm

Minimum cost is f(12.85)= pi*12.85^2*0.03+2*10000/12.85 *0.02=15.55+31.13=46.68

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