A space-probe fuel tank is to be constructed by attaching a hemispherical cap to the…

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A space-probe fuel tank is to be constructed by attaching a hemispherical cap to the top of a right circular cylindrical body. The bottom of the cylinder is closed by a flat circular plate made from the same alloy. The required total internal volume is

5000\,\text{cm}^3

. Let

r

(cm) denote the common interior radius of both the cylinder and hemisphere, and let

h

(cm) denote the interior height of the cylindrical section only. (a) Express the total exterior surface area

S(r,h)

of the tank (including the hemispherical cap and the flat bottom) in terms of

r

and

h

. (b) Write an equation expressing the fixed volume of

5000\,\text{cm}^3

in terms of

r

and

h

, and solve this equation for

h

in terms of

r

. (c) Substitute the expression from part (b) into

S(r,h)

to obtain

S

as a function of

r

only. Compute

\dfrac{dS}{dr}

and determine all critical values of

r

in the domain

r>0

. (d) Use the second-derivative test to confirm that the critical value found in part (c) corresponds to the minimum surface area. Then determine the corresponding value of

h

. (e) State the optimal tank dimensions and the least exterior surface area, both to the nearest

0.1\,\text{cm}

and

0.1\,\text{cm}^2

, respectively. Briefly explain why reducing surface area is advantageous for space missions.

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