A laboratory stores a cryogenic fluid in a cylindrical stainless-steel inner vessel…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A laboratory stores a cryogenic fluid in a cylindrical stainless-steel inner vessel of interior radius

r\ \text{cm}

and interior height

h\ \text{cm}

. To reduce heat transfer, the vessel is completely surrounded by a concentric layer of insulating foam of uniform thickness

t=1.0\ \text{cm}

. The interior volume available for the fluid must be

4000\ \text{cm}^3

. Stainless steel costs

0.035\ \text{dollars/cm}^2

of its exterior surface area. Foam insulation costs

0.008\ \text{dollars/cm}^2

of its exterior surface area, measured on the outside of the foam layer. Assume the top and bottom faces of both layers are flat and parallel, and ignore any gaps or seams. (a) Express the total cost

C(r,h)

of constructing one insulated vessel in terms of

r

and

h

. (b) Formulate the volume constraint for the interior space and solve for

h

in terms of

r

. (c) Substitute the expression for

h

into

C(r,h)

to obtain

C

as a function of

r

only. Compute

\frac{dC}{dr}

and find all critical values for

r>0

. (d) Prove that the critical value identified in part (c) yields the absolute minimum total cost by using an appropriate second-derivative argument or a sign chart for

\frac{dC}{dr}

. (e) The laboratory safety manual requires that the height-to-radius ratio of the inner vessel satisfy

1.2\le\frac{h}{r}\le3.0

for stability. Determine whether your optimal design from part (d) meets this safety requirement. If it does not, state which bound of the interval should be used instead and compute the resulting cost penalty to the nearest dollar.

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