Carlos wants to make a scale drawing of a rectangle. he wants the rectangle to have a length of (x-1) units and an area of 1-x/x^2+1 square units. What can you say about how wide the rectangle should be? Explain.
I will assume you want the area to be
(1-x)/(x^2 +1)
area = l x w
so (1-x)/(x^2 + 1 ) = w(x-1)
w = [ (1-x)/(x^2 + 1) ] / (x-1)
= -1/(x^2 + 1)
What I would say is "There is no real number for the width"
-since x^2 is always positive, then x^2 + 1 is always positive
Thus -1/(a positive) = a negative
We cannot have a negative width
unless you want your typed version to have a different meaning besides my interpretation.
the way you typed it it would have a value of 2 - 1/x
and I am pretty sure you didn't mean that.
To find out how wide the rectangle should be, we can use the given information about the length and area. Let's break down the problem step by step.
First, let's define the width of the rectangle as "w" units.
We are given that the length of the rectangle is (x-1) units. Therefore, the length (L) can be expressed as:
L = (x-1)
Also, we are given that the area of the rectangle is (1-x)/(x^2+1) square units. Therefore, the area (A) can be expressed as:
A = (1 - x) / (x^2 + 1)
Now, we know that the area of a rectangle is given by the formula:
A = L * w
Substituting the given values, we get:
(1 - x) / (x^2 +1) = (x - 1) * w
Now, to find out how wide the rectangle should be, we need to isolate "w" on one side of the equation. We can do this by cross-multiplying:
(1 - x) * w = (x - 1) * (x^2 +1)
Now, we can simplify this equation:
w - wx = x^3 - x + x^2 - 1
Rearranging the terms, we get:
w - wx = x^3 + x^2 - x - 1
Taking "x" as a common factor on the right-hand side, we can simplify further:
w - wx = x^2(x + 1) - (x + 1)
Factoring out (x + 1), we have:
w - wx = (x + 1)(x^2 - 1)
Now, we can divide both sides of the equation by (x - 1), as long as we assume that (x - 1) is not equal to zero. This is because division by zero is not defined:
(w - wx) / (x - 1) = (x + 1)(x^2 - 1) / (x - 1)
Dividing further, we get:
w = (x + 1)(x^2 - 1) / (x - 1)
Now, this expression gives us the width of the rectangle (w) in terms of the variable (x). We can simplify it further if necessary, but this is the general form of the equation for the width based on the given information.
So, in summary, the width of the rectangle should be (x + 1)(x^2 - 1) / (x - 1) units, based on the given length and area.