A closed cylindrical tank with radius $r$ m and height $h$ m is to be constructed to hold \(1000

Question

A closed cylindrical tank with radius $r$ m and height $h$ m is to be constructed to hold $1000$ m$^3$ of water. The material for the base and lid costs $50$ per m$^2$, while the material for the curved surface costs $30$ per m$^2$. Show that the total cost $C$ (in dollars) can be expressed as $C = 100\pi r^2 + \frac{60000}{r}$.[2] Use differentiation to find the exact radius of the cylinder for which $C$ is minimum. State the minimum cost.[4] Sketch the graph showing the total cost as the radius of the tank varies.[2] A change to one of the materials used results in the cylinder's volume doubling to $2000$ m$^3$, while maintaining the same minimum cost and radius found in part (b). State which material's cost has changed and its value.[1]

Accepted Answer

(b)$r=\sqrt[3]{\frac{300}{\pi}}$,$19690.29 (d)$15 per m$^2$

2025 YIJC P1 Q3

2025 YIJC P1 Q3