WEBVTT
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Find, to the nearest hundredth, the area of the circle π₯ plus nine squared plus π¦ minus three squared equals 49.0.
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The area of a circle can be found using the formula ππ squared, where π represents the radius of the circle.
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So in order to answer this question, we need to find the radius of the circle whose equation weβve been given.
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To do so, we can recall the center-radius form of the equation of a circle.
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If a circle has center with coordinates β, π and a radius of π units, then the center-radius form of its equation is π₯ minus β squared plus π¦ minus π squared equals π squared.
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We can see then that our circle is in center-radius form.
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And comparing the right side of the two equations, we have that π squared is equal to 49.0.
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Now, we havenβt actually been asked to give the coordinates of the center of the circle.
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But just for completeness, if we compare the two forms, we can see that π is equal to three because we have π¦ minus π in the second equation and π¦ minus three in the first.
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The value of β, we have to think about slightly more carefully because we have negative β in the general form but plus nine in our circle.
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So we have that negative β is equal to nine.
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And dividing through by negative one, we see that β is equal to negative nine.
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This means that the center of our circle is the point with coordinates negative nine, three.
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Although, as we already said, we havenβt actually been asked to work this out.
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Letβs return to calculating the radius of our circle.
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We have that π squared is equal to 49.0.
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So to find the value of π, we need to take the square root of both sides of the equation.
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Given π is equal to the square root of 49.0, which is exactly seven, we use the positive square root as the radius needs to be a positive value.
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Now, we didnβt actually need to perform the last steps and work out the value of π exactly because our formula for the area of the circle uses π squared not π.
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So once weβd determine that π squared is equal to 49.0 or just 49, we could in fact just substitute this value for π squared into our area formula.
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We have then that the area of this circle is equal to π multiplied by 49.
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Now, if we were asked to give an exact answer or if we didnβt have access to a calculator, we could leave our answer as a multiple of π.
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Itβs 49π.
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But in this question, weβve been asked to give our answer to the nearest hundredth.
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So we evaluate 49π on our calculator and it gives 153.93804, and the decimal continues.
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The digit in the next column to the right of the hundredth column, thatβs the thousandth column, is an eight.
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And as this is greater than five, this tells us that weβre rounding up.
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So the area of the circle, to the nearest hundredth, is 153.94.
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The units for this answer are just general square units.