Question Video: Finding the Perimeter of a Rectangle given Its Dimensions Involving Real Numbers Mathematics

Given that the dimensions of a rectangle are (57 + 7√2) cm and (57 − 7√2) cm, find the length of its perimeter.

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Video Transcript

Given that the dimensions of a rectangle are 57 plus seven root two and 57 minus seven root two centimeters, find the length of its perimeter.

So I’ve drawn a sketch of our rectangle. And it’s the perimeter that we’re looking to find. And the perimeter is the distance all the way around our rectangle. So to help us do that, what I’m gonna do is I’m actually going to write our values for our dimensions onto our diagram. So first of all, we have our length which is 57 plus seven root two and then we’ll have our width which is 57 minus seven root two.

Now that we’ve actually marked our dimensions, what we can now use is our formula here to find the perimeter. The perimeter is equal to two 𝑙 plus two 𝑤. So it’s two multiplied by the length plus two multiplied by the width. So therefore, to find our perimeter, all we need to do is substitute our length which is 57 plus seven root two and our width 57 minus seven root two into our formula. So we’re gonna get 𝑃. So our perimeter is equal to two multiplied by 57 plus seven root two plus two multiplied by 57 minus seven root two.

So now we’re going expand our parenthesis. So first of all, I have two multiplied by 57, which gives us 114. And then, this is plus 14 root two because you have two multiplied by seven root two. So two multiplied by seven is 14. So that’s 14 root two. So then moving on to our second parenthesis, we have two multiplied by 57 again. So again, we get plus 114. And then, the final multiplication is two multiplied by this time negative seven root two. So therefore, we get minus 14 root two. So we now have an expression for our perimeter which is perimeter is equal to 114 plus 14 root two plus 114 minus 14 root two.

So now all we have to do is simplify to find our perimeter. So therefore, we reach our final answer. And we can say that given the dimensions of a rectangle are 57 plus seven root two and 57 minus seven root two, the length of its perimeter is equal to 228 centimeters. And we got that because we had 114 plus 114 which is 228 and then positive 14 root two minus 14 root two which is just zero. So that left us with our final answer of 𝑃 is equal to 228 centimeters.

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