A closed right circular cylindrical storage tank must hold of liquid

MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A closed right circular cylindrical storage tank must hold

1000\,\text{m}^3

of liquid. During fabrication, an additional

10\%

of the curved-surface area is required as extra material for welding allowances. The flat ends are made from steel costing Rs

120

per m

^{2}

, while the curved surface (including the extra

10\%

allowance) costs Rs

100

per m

^{2}

. (i) Let

r\,\text{m}

be the base radius of the tank. Express the total material cost

C(r)

in terms of

r

alone. (ii) Obtain the value of

r

that minimises

C(r)

and determine the corresponding height of the tank. (iii) Using the Second Derivative Test, confirm that the critical point found provides the least cost. (iv) Compute the minimum total material cost of the tank (give your answer correct to the nearest rupee).

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