WEBVTT
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Write an expression for the area of the shaded region in the shape below.
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We have a diagram of a rectangle with three shapes cut out of it, and the remaining area is what weβre being asked to calculate.
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The dimensions of the rectangle and the cut-out shapes are given in terms of the letter π€, which is why weβre asked to write an expression for the area rather than calculate it.
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Letβs think about how to approach this question.
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In order to find the shaded area, we first need to find the area of the larger rectangle.
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We then need to subtract the areas of each of the cut-out shapes.
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So we have four areas, for which we need to find expressions all together.
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Letβs look at the area of the larger rectangle first of all.
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To find the area of a rectangle, we multiply the length by the width.
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In this case, the measurements are 13 π€ and 11 π€ plus 13.
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The result of multiplying these two expressions together is 13 π€ into 11 π€ plus 13.
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And weβll expand this bracket later on.
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Now letβs look at the areas of the cut-out shapes.
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There is a rectangle at the top of the diagram, which has dimensions of two and three π€.
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Its area is therefore two multiplied by three π€, which is six π€.
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The other two cut-out shapes are two identical squares, both with sides of length three π€.
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The areas of each of these squares are found by multiplying three π€ by three π€, which gives a result of nine π€ squared.
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Now, letβs substitute these expressions for the area of the cut-out shapes into our expression for the shaded area.
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We have 13 π€ multiplied by 11 π€ plus 13 as before minus six π€ and then minus nine π€ squared minus nine π€ squared again.
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Remember there were two of these squares that we need to subtract.
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Next, we need to simplify our expression.
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So weβll begin by expanding the brackets.
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The result of expanding this bracket is 143 π€ squared plus 169 π€.
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Weβre still subtracting six π€ and simplifying minus nine π€ squared minus another nine π€ squared.
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We now have minus 18 π€ squared.
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The final step is to simplify our expression by grouping the like terms.
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143 π€ squared minus 18 π€ squared gives 125 π€ squared and positive 169 π€ minus six π€ gives positive 163 π€.
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And so we have our expression for the area of the shaded region: 125 π€ squared plus 163 π€.