2021 GMC 12B Problems/Problem 13

Problem

Let $a$ and $b$ be two legs of a right triangle with hypotenuse $2$ . Find the greatest possible value of $a^3+a^2b+ab^2+b^3$

$\textbf{(A)} ~4\sqrt{2}+2 \qquad\textbf{(B)} ~4\sqrt{3}+4 \qquad\textbf{(C)} ~8\sqrt{2} \qquad\textbf{(D)} ~6\sqrt{3} \qquad\textbf{(E)} ~12$

Solution

To maximize $a^3+a^2b+ab^2+b^3$ is to maximize $a$ and $b$ . Thus, $b = a$ . Applying the Pythagorean Theorem, we get $\begin{align*} a^2 + b^2 &= 4 \\ 2a^2 &= 4 \\ a^2 &= 2 \\ a &= \sqrt2. \end{align*}$

Evaluating $a^3+a^2b+ab^2+b^3$ gives $\boxed{\textbf{(C)}~8\sqrt2}$ .

~pineconee

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2021 GMC 12B Problems/Problem 13

Problem

Solution