Mathematics

Problem

The graph of

y=6-x^{2}

(for

x\ge 0

) and the

x

-axis enclose a region that is cut off on the right by the vertical line

x=k

, where

0<k<3

. For what value of

k

is the area of this region greatest?

Intuitive explanation

You're optimizing the area of a region bounded by a parabola, the $x$ -axis, and a movable vertical line. The key insight is that as you slide the line rightward from $x=0$ , you're adding area at first, but the parabola dips lower and lower, so eventually you start removing more than you gain. Set up the area as a function of $k$ , then find where its derivative equals zero—that's your maximum.

Intuitive explanation

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