WEBVTT
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In the figure, πππ
, π
ππ, and ππ΄π a three semicircles of diameters 10 centimeters, three centimeters, and seven centimeters, respectively.
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Find the perimeter of the shaded region.
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Take π is equal to 3.14.
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For this question, weβve been given a shape and weβve been asked to find its perimeter.
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The perimeter of the shape that weβve been given is comprised of three semicircular arcs.
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If we define the perimeter as π₯, we can say that π₯ is the sum of these three arcs.
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The arcs are πππ
, π
ππ, and ππ΄π.
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Now, the question tells us the diameter of the semicircles which form these arcs.
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This information can be used to answer the question in the following way.
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We first recall that the perimeter β also known as the circumference π of a circle β is given by two ππ, where π is the radius of the circle.
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Since the semicircle is a circle thatβs been cut in half, the arc length of a semicircle will be half that of the circumference of the original circle; that is, π divided by two.
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Dividing both sides of our equation by two, we see that this is equal to two ππ over two.
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Cancelling the two on the top and bottom half of our fraction, we see that this is equal to π times π.
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Now, for this question, we havenβt been given the radius of the circles, which have formed our semicircular arcs rather weβve been given the diameters.
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We can now recall that the radius of a circle is half of its diameter.
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We can now put these two factors together by replacing the π in our original formula by π over two.
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When we do so, we find that the length of a semicircular arc, here given by π over two, is equal to π times the diameter divided by two.
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This formula will allow us to move forward with our question.
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And weβll put it to one side to make room for some working.
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Looking back at the equation that we have for π₯, which we defined as the perimeter of the shaded region, we can now replace the length πππ
with π times the diameter of the semicircle which formed πππ
divided by two.
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We can also do the same thing for the semicircles π
ππ and ππ΄π.
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The question has given us that the diameter of πππ
is 10 centimeters.
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And so we can rewrite our first term as π times 10 over two.
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The diameter of π
ππ is three centimeters.
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And so our second term becomes π times three over two.
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Finally, the diameter of ππ΄π is seven centimeters.
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And our last term becomes π times seven over two.
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We can now use the fact that all three of our terms have a factor of π.
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We can take out this factor to simplify our next line of working, which then becomes π times 10 over two plus three over two plus seven over two.
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10 plus three plus seven is equal to 20.
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And so our bracket simplify to 20 over two.
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20 divided by two is 10.
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And so we can replace this in our brackets.
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Weβre now left with the fact that π₯, the perimeter of our shape, is equal to π times 10.
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Here, we know that the question has given us an approximate value of π to use, which is 3.14.
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We now substitute this value into our equation.
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3.14 times 10 is equal to 31.4.
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And we also remember to add back in the units of length, which here are centimeters.
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This is now our final answer for the question.
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And we have found that based on the semicircular diameters which weβve been given the perimeter of the shaded region is 31.4 centimeters.