Challenge: Amy is laying tiles in her rectangular bathroom. By the time she has finished, she has used [tex]$8 m^2$[/tex] worth of tiles. She knows the length of one side of the room is [tex]$(\sqrt{5}+2) m$[/tex] but, unfortunately, she has lost her tape measure. Amy still needs to work out the perimeter of the room. Calculate the perimeter of the room, giving your answer in its simplest form.
Answer (2)
To find the perimeter of Amy's bathroom, we first determined the width by dividing the area by the length. After rationalizing the denominator, we calculated the perimeter using the formula P = 2 ( l + w ) , resulting in P = 18 5 − 28 m .
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Calculate the width of the rectangle by dividing the area by the given length: w = 5 + 2 8 .
Rationalize the denominator of the width expression: w = 8 ( 5 − 2 ) .
Calculate the perimeter using the formula P = 2 ( l + w ) , substituting the values of l and w .
Simplify the expression to find the perimeter: P = 18 5 − 28 . The perimeter of the room is 18 5 − 28 .
Explanation
Problem Analysis Let's analyze the problem. We know the area of the rectangular bathroom is 8 m 2 and the length of one side is ( 5 + 2 ) m . We need to find the perimeter of the room.
Finding the Width Let A be the area of the rectangle, l be the length, and w be the width. We have A = 8 m 2 and l = ( 5 + 2 ) m . The area of a rectangle is given by A = l × w . Therefore, we can find the width w by dividing the area by the length: w = l A = 5 + 2 8
Rationalizing the Denominator To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is ( 5 − 2 ) : w = 5 + 2 8 × 5 − 2 5 − 2 = 5 − 4 8 ( 5 − 2 ) = 8 ( 5 − 2 ) So, the width is w = 8 ( 5 − 2 ) m .
Calculating the Perimeter Now that we have the length and width, we can find the perimeter P using the formula P = 2 ( l + w ) . Substituting the values of l and w , we get: P = 2 (( 5 + 2 ) + 8 ( 5 − 2 )) = 2 ( 5 + 2 + 8 5 − 16 ) = 2 ( 9 5 − 14 ) Simplifying the expression, we have: P = 18 5 − 28
Final Answer Therefore, the perimeter of the room is ( 18 5 − 28 ) m .
Examples
Imagine you're designing a rectangular garden and need to calculate the amount of fencing required. If you know the area of the garden and the length of one side, you can use the same method to find the other side and then calculate the perimeter, which tells you how much fencing you need. This is a practical application of area and perimeter calculations in real-world scenarios.
