Wheels of Fury - Hidden Object

"Unleash the Fury of Wheels - Hidden Object: Assist entitled Eddie in rectifying the chaos he has caused in his room. Highlights: 10 lively stages, Zoom in to get a closer look, 40 items per level, 400 hidden objects to uncover, Bonus system in place."<|endoftext|><|endoftext|> # 2006 AMC 12B Problems/Problem 6. (Redirected from 2006 AMC 12B Problem 6) ## Problem. A circle of radius $r$ is inscribed in a corner of a rectangle as shown. The ratio of the length of the rectangle to its width is $3:4$. What is the value of $r$? [asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(0,0), B=(4,0), C=(4,3), D=(0,3), O=(1,1); pair[] dotted={A,B,C,D,O}; draw(A--B--C--D--cycle); draw(Circle(O,1)); dot(dotted); label("$r$",O,SE); [/asy] $\mathrm{(A)}\ \frac{3}{8}\qquad\mathrm{(B)}\ \frac{7}{16}\qquad\mathrm{(C)}\ \frac{1}{2}\qquad\mathrm{(D)}\ \frac{5}{8}\qquad\mathrm{(E)}\ \frac{2}{3}$ ## Solution. Let the length of the rectangle be $3x$ and the width be $4x$. The diameter of the circle is the length of the rectangle, so it is $3x$. The radius of the circle is half the diameter, so it is $\frac{3x}{2}$. The area of the circle is $\pi r^2$, so it is $\frac{9\pi x^2}{4}$. The area of the rectangle is $3x \cdot 4x = 12x^2$. The area of the circle is $\frac{9\pi x^2}{4}$, so the ratio of the area of the circle to the area of the rectangle is $\frac{\frac{9\pi x^2}{4}}{12x^2} = \frac{3\pi}{16}$. The ratio of the area of the circle to the area of the rectangle is also equal to the ratio of the length of the rectangle to the width, which is $\frac{3}{4}$. Therefore, $\frac{3\pi}{16} = \frac{3}{4}$, so $\pi = \frac{4}{16} = \frac{1}{4}$. The radius of the circle is $\frac{3x}{2}$, so $\frac{3x}{