A beverage company must redesign its standard aluminum can so that it uses exactly…
MATHEMATICS · Multi-part · CBSE · Mathematics · Surface Areas and Volumes · Surface Areas and Volumes
Problem
A beverage company must redesign its standard aluminum can so that it uses exactly 180 cm2 of material for the combined lateral surface and the two circular ends. The can will still be a right circular cylinder and must store liquid.
Throughout this question, let r denote the radius (in cm) of the circular base and let h denote the height (in cm) of the can.
(a) Describe, using words and symbolic relationships, why r and h cannot vary independently when the total amount of aluminum is fixed at 180 cm2. Identify the geometric quantities that link the two variables.
(b) Derive an explicit formula for the internal volume V of the can in terms of r alone. Clearly state the algebraic steps required to obtain this single-variable expression.
(c) State the domain of r that is physically meaningful for the redesigned can, and explain how the geometric context justifies each endpoint of the interval.