MathDB

18

Part of 2012 AMC 10

Problems(2)

Area of a Closed Curve

Source: 2012 AMC10A Problem #18

2/9/2012
The closed curve in the figure is made up of 99 congruent circular arcs each of length 2π3\frac{2\pi}{3}, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 22. What is the area enclosed by the curve?
[asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("∘\circ",(0,1)); label("∘\circ",(0.865,0.5)); label("∘\circ",(-0.865,0.5)); label("∘\circ",(0.865,-0.5)); label("∘\circ",(-0.865,-0.5)); label("∘\circ",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy]
<spanclass=′latex−bold′>(A)</span> 2π+6<spanclass=′latex−bold′>(B)</span> 2π+43<spanclass=′latex−bold′>(C)</span> 3π+4<spanclass=′latex−bold′>(D)</span> 2π+33+2<spanclass=′latex−bold′>(E)</span> π+63 <span class='latex-bold'>(A)</span>\ 2\pi+6\qquad<span class='latex-bold'>(B)</span>\ 2\pi+4\sqrt3 \qquad<span class='latex-bold'>(C)</span>\ 3\pi+4 \qquad<span class='latex-bold'>(D)</span>\ 2\pi+3\sqrt3+2 \qquad<span class='latex-bold'>(E)</span>\ \pi+6\sqrt3
geometryAMC 10AMC
Disease and Blood Test

Source: 2012 AMC10B Problem #18

2/23/2012
Suppose that one of every 500500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a 2%2\% false positive rate; in other words, for such people, 98%98\% of the time the test will turn out negative, but 2%2\% of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let pp be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to pp?
<spanclass=′latex−bold′>(A)</span> 198<spanclass=′latex−bold′>(B)</span> 19<spanclass=′latex−bold′>(C)</span> 111<spanclass=′latex−bold′>(D)</span> 4999<spanclass=′latex−bold′>(E)</span> 9899 <span class='latex-bold'>(A)</span>\ \frac{1}{98}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{9}\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{11}\qquad<span class='latex-bold'>(D)</span>\ \frac{49}{99}\qquad<span class='latex-bold'>(E)</span>\ \frac{98}{99}
probabilityconditional probabilityexpected valueAMC