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Difference between revisions of "2021 GMC 12B Problems/Problem 13"

(Created page with "==Problem== Let <math>a</math> and <math>b</math> be two legs of a right triangle with hypotenuse <math>2</math>. Find the greatest possible value of <math>a^3+a^2b+ab^2+b^3</...")
 
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Latest revision as of 18:59, 6 March 2022

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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