SOLUTION: a rectangular room has an area of 665 square feet. Its length is three less than twice its width. what are the dimensions of the room?

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Question 211852: a rectangular room has an area of 665 square feet. Its length is three less than twice its width. what are the dimensions of the room?
Answer by drj(1380) About Me  (Show Source):
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A rectangular room has an area of 665 square feet. Its length is three less than twice its width. what are the dimensions of the room?

Step 1. Let l = 2w-3 length of rectangle and let w be the width

Step 2. Let A = 655 square feet be the Area = (length)(width)

A=655=%282w-3%29w

655=2w%5E2-3w

Step 3. Subtract 655 to both sides of equation to get 0 by left side

655-655=2w%5E2-3w-655

0=2w%5E2-3w-655

Step 4. Now we have a quadratic equation so we can use the quadratic formula given as

w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=2, b=-3 and c=-655.
The following steps solve the quadratic equation:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 2w%5E2%2B-3w%2B-655+=+0) has the following solutons:

w%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A2%2A-655=5249.

Discriminant d=5249 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+5249+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+5249+%29%29%2F2%5C2+=+18.86249568668
w%5B2%5D+=+%28-%28-3%29-sqrt%28+5249+%29%29%2F2%5C2+=+-17.36249568668

Quadratic expression 2w%5E2%2B-3w%2B-655 can be factored:
2w%5E2%2B-3w%2B-655+=+2%28w-18.86249568668%29%2A%28w--17.36249568668%29
Again, the answer is: 18.86249568668, -17.36249568668. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-3%2Ax%2B-655+%29



Step 5. Choose the positive number for a positive length. In this case, the width is 18.86 ft and length is 2w+3 or 2(18.86)-3 or 34.72 ft.

Step 6. Answer is 18.86 ft for the width and 34.72 for the length. Multiply these two numbers to get an area of 655 square feet.

Hope the above steps were helpful. Good luck in your homework and studies!

Respectfully
Dr J

Hope you understood and followed the steps. Good luck in your studies. Dr J

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