WEBVTT
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A target board consists of three concentric circles of radii six centimetres, 12 centimetres, and 24 centimetres, which define three regions, gold, red, and blue as shown in the diagram.
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If a dart is shown randomly such that it hits the target, what is the probability it will hit the red region?
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In order to answer this question, we need to calculate the area of the three regions.
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We recall that the area of a circle is equal to π multiplied by the radius squared.
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The gold circle has a radius of six centimetres.
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This means that the area of the target board that is gold is equal to π multiplied by six squared.
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Six squared is equal to 36.
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Therefore, the gold area is equal to 36π.
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Weβre told that the red circle has a radius of 12 centimetres.
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The area of the target board that is red will be equal to the red circle minus the gold circle.
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This is equal to π multiplied by 12 squared minus π multiplied by six squared.
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12 squared is equal to 144.
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Therefore, the area of the red circle is 144π.
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We already know that the area of the gold circle is 36π.
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144 minus 36 is equal to 108.
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Therefore, the red area is 108π.
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We can repeat this process for the blue area.
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The area of the blue circle is equal to π multiplied by 24 squared.
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And we need to subtract π multiplied by 12 squared.
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24 squared is equal to 576.
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This means we need to subtract 144π from 576π.
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This is equal to 432π.
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Dividing all three of these values by π tells us that the ratio of gold to red to blue is 36 to 108 to 432.
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All three of these numbers are divisible by 36.
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36 divided by 36 is equal to one.
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108 divided by 36 is equal to three.
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And 432 divided by 36 is equal to 12.
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The ratio of the gold area to the red area to the blue area simplifies to one to three to 12.
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We were asked to calculate the probability that a dart hits the red region.
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The red region is equal to three parts of our ratio.
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We have a total of 16 parts, as one plus three plus 12 is equal to 16.
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This means that the probability that the dart hits the red region is three out of 16 or three sixteenths.