Finding the dimensions of a rectangle given the area and diagonal

A

Am0stCr0ok3d

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Hello everyone,

I'm in need of some help. I have a question on my assignment that gives me the area (180) and diagonal (20) of a rectangle and I have to find the length and width from this. So far I have 2 equations, Area = LW and L^2+ W^2 = c^2. So, 180 = LW, solve for W, W = 180/L and then L^2 + W^2 = 20^2, which is L^2 + (180/L)^2 = 400. Then I've done, 400 = L^2 + (32400/L^2). That's as far as my brain will take me. I know it will turn into a quadratic but I've spent so much time on it that my brain has just shut down lol

May someone please help me understand this?
 
Am0stCr0ok3d said:
Hello everyone,

I'm in need of some help. I have a question on my assignment that gives me the area (180) and diagonal (20) of a rectangle and I have to find the length and width from this. So far I have 2 equations, Area = LW and L^2+ W^2 = c^2. So, 180 = LW, solve for W, W = 180/L and then L^2 + W^2 = 20^2, which is L^2 + (180/L)^2 = 400. Then I've done, 400 = L^2 + (32400/L^2). That's as far as my brain will take me. I know it will turn into a quadratic but I've spent so much time on it that my brain has just shut down lol

May someone please help me understand this?
Click to expand...
Assume:

u = L^2.....then...

400 = u + 32400/u

u^2 - 400 * u + 32400 = 0

Now you have a quadratic - and solve.
 
Is the reason for the change in a variable to make the equation easier to follow and therefore solve? Also, the discriminant of that will give me 2 positive answers. I'm not sure what that means in terms of having 2 different lengths. Is that because there are 2 possible lots of dimensions, landscape and portrait kind of thing?

Thank you for your prompt response also! Very much appreciate it.
 
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Am0stCr0ok3d said:
View attachment 26958

This is my current working out.
Click to expand...
Excellent work! I specially appreciate the neat presentation. That habit (neat presentation) will serve you well.

As far as the two answers, yes. If you had worked a little further and calculated 'L' (= √u) and 'W' (=180/L) from those two values, you would seen that the value 'L' from one answer, becomes the value of 'W' from the other answer. - hence 'Landscape' ↔ 'Portrait'.

Again - excellent work.
 
Am0stCr0ok3d, there are a few corrections regarding that grid paper you wrote on.

On the section with the circled "2," it is correct down to and including the fourth line.
The fifth line should not exist, because \(\displaystyle \ L^4 \ \) is incorrect/not part of a step.

Below that you are going to let \(\displaystyle \ x = L^2, \ \) so the next line should be

\(\displaystyle 400 = x + \dfrac{32400}{x} \)

Multiply both sides by x:

\(\displaystyle 400x = x^2 + 32400 \)

\(\displaystyle x^2 - 400x + 32400 = 0 \)
 
Last edited:
lookagain said:
Am0stCr0ok3d, there are a few corrections regarding that grid paper you wrote on.

On the section with the circled "2," it is correct down to and including the fourth line.
The fifth line should not exist, because \(\displaystyle \ L^4 \ \) is incorrect/not part of a step.

Below that you are going to let \(\displaystyle \ x = L^2, \ \) so the next line should be

\(\displaystyle 400 = x + \dfrac{32400}{x} \)

Multiply both sides by x:

\(\displaystyle 400x = x^2 + 32400 \)

\(\displaystyle x^2 - 400x + 32400 = 0 \)
Click to expand...
I see, I'll delete that from my notes. I did change it after the "Assume \(\displaystyle \ x = L^2 \ \)" part. I'll remove it just to be safe with my assignment marks etc. Thankyou!!
 
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