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A certain rectangle has an area of 48 and a perimeter of 32. What is

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Bunuel
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A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink] New post   13 Feb 2023, 23:51
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A certain rectangle has an area of 48 and a perimeter of 32. What is the length of the diagonal of the rectangle?

A. 4
B. \(4\sqrt{3}\)
C. 16
D. \(4\sqrt{10}\)
E. 160
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D
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Re: A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink] New post   14 Feb 2023, 07:43
Bunuel wrote:
A certain rectangle has an area of 48 and a perimeter of 32. What is the length of the diagonal of the rectangle?

A. 4
B. \(4\sqrt{3}\)
C. 16
D. \(4\sqrt{10}\)
E. 160

\(lb = 48\) ; \(2(l +b) = 32\)

So, \(l + b = 16\)

Or, \(l = 16 - b\)

Thus, \(lb = 48 = (16 - b)b\)

Or, \(16b - b^2 = 48\)

Or, \(b^2 - 16b + 48 = 0\)

Or, \(b= 12 , 4\)

If \(b = 4\) ; \(l = 12\) (Possible as \(l > b\))
If \(b = 12\) ; \(l = 4\) (Not Possible as \(l < b\))

Thus, Diagonal will be \(\sqrt{4^2 + 12^2} = \sqrt{160}\) = \(4\sqrt{10}\), IMHO (D)
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Re: A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink] New post   17 Feb 2023, 07:41
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Sorry how is the answer E?
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Bunuel
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Re: A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink] New post   17 Feb 2023, 09:11
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bhavik29 wrote:
Sorry how is the answer E?


The OA is D. Edited. Thank you for noticing!
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Re: A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink] New post   17 Feb 2023, 13:12
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Bunuel wrote:
A certain rectangle has an area of 48 and a perimeter of 32. What is the length of the diagonal of the rectangle?

A. 4
B. \(4\sqrt{3}\)
C. 16
D. \(4\sqrt{10}\)
E. 160


We can create the equations where L = Length and W = Width.
LW = 48

And

2L + 2W = 32

L + W = 16

If we square L + W = 16, we have:

L^2 + W^2 + 2LW = 16^2

L^2 + W^2 + 2LW = 256

Since LW = 48, we have:

L^2 + W^2 + 96 = 256

L^2 + W^2 = 160

Lastly, since diagonal^2 = L^2 + W^2, we have:

160 = D^2

√16 x √10 = D

4√10 = D

Answer: D
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Re: A certain rectangle has an area of 48 and a perimeter of 32. What is [#permalink]
17 Feb 2023, 13:12
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