Answer:
L= 11 m and W = 2 m
Step-by-step explanation:
Let's convert the sentences into mathematical expressions, for which we need to assign letters to the unknowns: length and width of a rectangle.
Let's identify the rectangle's length with the letter "L', and its width with the letter "W".
Then we can write:
"The length of a rectangle is 3 meters more than 4 times the width."
as: L = 4 * W + 3
The second sentence says:
"The area (of the rectangle) is 22 square meters", so to write this in mathematical terms we need to recall the formula for the area of a rectangle:
Area = Length * Width (the product of the rectangle's length times its width)
Therefore the second sentence can be converted into the following equation:
Area = L * W = 22
Now we can use the first mathematical expression we constructed in the second formula so we can reduce the number of unknowns from two (L and W) to only one (W) and solve for it in the equation.
We replace "L" with 4 * W + 3 (from our first expression) in the area formula:
L * W = (4 * W + 3) * W = 22
[tex]4W^2 + 3W = 22\\4W^2+3W-22=0[/tex]
Which is a quadratic equation in W, and which has solutions given by the quadratic formula:
[tex]ax^2+bx+c=0\\[/tex]
[tex]x=\frac{-b+/- \sqrt{b^2-4ac} }{2a}[/tex]
For our unknown W (instead of "x") and our parameters:
[tex]a=4, b=3, c=-22[/tex]
the quadratic formula would be:
[tex]W=\frac{-3+/- \sqrt{3^2-4(4)(-22)} }{2(4)}=\frac{-3+/- \sqrt{9+352} }{8}= \frac{-3+/- \sqrt{361} }{8}=\frac{-3+/-19 }{8}[/tex]This gives as two possible solutions (one using the plus and the other using the minus):
[tex]W=\frac{-3-19}{8} =\frac{-22}{8}=-\frac{11}{4} \\W=\frac{-3+19}{8} =\frac{16}{8}=2 \\[/tex]
the negative answer has no physical meaning in our case because we are dealing with positive dimensions for a rectangle. Therefore the only logical solution for W is: 2 meters
Now we use the first expression we found for "L" to find its value:
L = 4 (2) +3 = 8 + 3 = 11 meters
