Scaling Pizza Size for More People

This question uses a pizza-sharing scenario to test whether candidates understand how quantities that depend on area scale when the underlying length changes. Instead of treating "number of people served" as a linear function of diameter, the problem forces you to recognize that the relevant physical quantity is area, which grows faster than the diameter itself. The second part explicitly asks you to link this scaling idea to derivatives, so you must move beyond plug-and-chug geometry and articulate how infinitesimal changes in size affect area-based outcomes.

Conceptually, the problem leans on area formulas, proportional reasoning, and recognizing quadratic dependence on a linear variable. Mathematically, it invites discussion of how a function behaves under scaling, and how the derivative encodes sensitivity of area to changes in radius or diameter. Interviewers watch for candidates who clearly distinguish between linear and quadratic growth, translate a word problem into functional relationships, and then justify the connection to derivatives in terms of rates of change rather than formula memorization.

What it tests

Whenever a quantity depends on the area of a shape, and the area itself depends on the square of a linear dimension (like diameter or radius), scaling up the area by a factor $k$ requires scaling the linear dimension by $\sqrt{k}$. This is because area is a quadratic function of the linear measure: if $A \propto d^2$, then to achieve $A_2 = k A_1$, you need $d_2 = d_1 \sqrt{k}$. This pattern appears in any context where the effect (serving size, probability, payout) is determined by a squared or higher-power relationship to a base variable. The reason this holds is that multiplying the base variable by $\sqrt{k}$ squares to give the desired scaling in area or variance, preserving proportionality.