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Given that the area of the coloured part in the figure is 155 square foot, find the value of ๐ฅ to the nearest tenth.
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Letโs recall what we actually remember about the area of a circular sector.
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For a circular sector with a radius ๐ and an angle ๐ radians, its area can be found using the formula, a half ๐ squared ๐.
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We actually know that the area of our circular sector is 155 square feet.
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And weโre looking to find the length of the radius.
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Weโre going to substitute what we know about our sector into the formula for sector area.
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However, before we do, we need to convert the angle 239 degrees into radians.
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To do this, we recall that two ๐ radians is equal to 360 degrees.
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We can then divide through by 360.
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And that tells us that one degree is equal to two ๐ divided by 360, which is equal to ๐ by 180 radians.
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Once we have this, we can multiply through by 239.
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And that tells us that 239 degrees is equal to 239๐ over 180 radians.
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Substituting what we now know about our circular sector into the formula for the area tells us that the area is a half ๐ฅ squared multiplied by 239๐ over 180.
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And of course, we know that the area is 155.
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So we equate this expression to 155.
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At this point, we could multiply by two and then by 180 to begin solving the equation.
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Instead though, we choose to do that in one step and multiply both sides by 360.
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155 multiplied by 360 is 55800.
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So we see that 55800 is equal to 239๐๐ฅ squared.
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We then divide through by 239๐.
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That gives us 74.316 and so on is equal to ๐ฅ squared.
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Now we wonโt round this number just yet.
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Instead to solve, weโre going to find the square root of both sides of our equation.
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So weโll square root this number in its unrounded form.
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Thatโs 8.6207 and so no.
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Now, when we square root, we should recall that there are two solutions, a positive and a negative.
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In this case though, we can immediately disregard the negative solution as ๐ฅ is a length.
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And we canโt have a negative length.
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Finally, we round this number to the nearest tenth.
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And that tells us that ๐ฅ is equal to 8.6.
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We donโt need to give any units here since weโre told that ๐ฅ is the number of feet.
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So ๐ฅ is simply a number.
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It is 8.6.