Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.3 AMC 10/12 Find the answer

Let $K$ be the measure of the area bounded by the $x$ -axis, the line $x=8$ , and the curve defined by
$f={(x,y)\quad |\quad y=x \text{ when } 0 \le x \le 5, y=2x-5 \text{ when } 5 \le x \le 8}.$
Then $K$ is:

Pick one

Solution

The shape can be divided into a triangle and a trapezoid. For the triangle, the base is $5$ and the height is $5$ , so the area is $\frac{5 \cdot 5}{2} = \frac{25}{2}$ . For the trapezoid, the two bases are $5$ and $11$ and the height is $3$ , so the area is $\frac{3(5+11)}{2} = 24$ . Thus, the total area is $\frac{25}{2} + 24 = \frac{73}{2} = \boxed{\textbf{(C) } 36.5}$ .

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