Pharmaceutical capsules are sometimes modeled as a right circular cylinder of length…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Pharmaceutical capsules are sometimes modeled as a right circular cylinder of length h capped on each end by identical hemispheres of radius r. A particular medication requires each capsule to enclose 2.5 cm3 of volume. The capsule body (the cylindrical portion) is molded from material costing 0.03percm^{2},whilethehemisphericalendcapsaremoldedfromaspecializedmaterialcosting0.06 per cm2. The manufacturer seeks to minimize the total material cost per capsule.
(a) Express the total interior volume of the capsule in terms of r and h, and impose the fixed‐volume condition.
(b) Derive a formula for the total material cost K (in dollars) in terms of r and h.
(c) Use the volume constraint to rewrite K as a function of a single variable. Clearly identify the domain that is physically meaningful.
(d) Determine the radius and cylindrical length that minimize the material cost. Then find the minimum cost, to the nearest tenth of a cent.
(e) The specialized material used for the hemispherical caps has recently become more expensive. Describe qualitatively how an increase in the cap material cost would affect the optimal ratio of h to r, and justify your reasoning.