Question

4.6 optimization #28

a cylindrical container with no top is to be constructed to hold a fixed volume V_{0 }of liquid. the cost of the material used for bottom is $0.50/in^{2 }and cost of the material used for the curved lateral side is $0.30/in^{2 }.

use calculus to find radius (in terms of V_{0}) of the leaset esxpensive container.

Step 1

Consider that the radius of the cylindrical container with no top is r, and the height is h. The volume becomes V_{0} = πr^{2}h. The area of the bottom is A_{1} = πr^{2} and the area of the curved lateral side is A_{2} = 2πrh. The material costs for the bottom and the curved lateral side are $0.50/in^{2} and $0.30/in^{2}, respectively. The cost function for the total material used becomes

Step 2

As V_{0} is fixed, h = V_{0}/πr^{2}, thus the modified cost function becomes

Step 3

Differentiate A with respe...

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