Problem

Use the cylinder diagram and \(V~=~\pi\cdot~r^2\cdot~h\) with fixed volume \(500\). Solve for \(h\) as a function of \(r\), state the feasible domain and range, and describe the graph behavior and asymptotes.

Positive coordinate scaffold for graphing height h versus radius r; the solved relationship and curve are withheld.

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What this problem is really about

Substitute the fixed volume and isolate the variable assigned to the vertical axis before thinking about the graph. Positive cylinder dimensions retain only the first-quadrant branch. Then use the squared radius in the denominator to predict decreasing behavior and analyze both boundary directions, distinguishing values the curve approaches from dimensions a physical cylinder can actually attain.

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h=500πr², with r>0 and h>0. Its graph is an increasing quadratic branch from (0,0).

h=500/(πr²), with r>0 and h>0. Its graph is a decreasing inverse-square curve with vertical asymptote r=0 and horizontal asymptote h=0.

h=500/(πr²), with r≠0. Its graph has two inverse-square branches.

h=500/(πr), with r>0 and h>0. Its graph is a decreasing reciprocal curve.

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Use the cylinder diagram and \(V~=~\pi\cdot~r^2\cdot~h\) with fixed volume \(500\). Solve for \(h\) as a function of \(r\), state the feasible domain and range, and describe the graph behavior and asymptotes.

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