a farmer has a field with area of x^2+2x-15 meters and a length of x^2+3x-18 over x+1 meters. what is the width of the field? simplify and state restrictions. Recall: a=l x w
To find the width of the field, we divide the area of the field by its length:
Width = Area / Length
The area of the field is given by x^2+2x-15 meters, and the length is given by (x^2+3x-18)/(x+1) meters.
So, Width = (x^2+2x-15) / (x^2+3x-18)/(x+1)
To divide by a fraction, we can multiply by its reciprocal:
Width = (x^2+2x-15) * (x+1) / (x^2+3x-18)
Expanding both the numerator and denominator:
Width = (x^3 + x^2 + 2x^2 + 2x - 15x - 15) / (x^2 + 3x - 18)
Combining like terms in the numerator:
Width = (x^3 + 3x^2 - 13x - 15) / (x^2 + 3x - 18)
Now, let's simplify this expression further by using long division:
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x+3 | x^3 + 3x^2 -13x -15
- (x^3 + 3x^2 - 18x)
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5x + 3
Therefore, we have:
Width = x + 3 + (5x+3) / (x+3)
Simplifying, we get:
Width = x + 5 - (5/ (x+3))
The width of the field is x + 5 - (5/ (x+3)) meters.
Restrictions:
We need to consider restrictions that might result in division by zero. In this case, if x+3 equals zero, the divisor will become zero which is undefined. Therefore, x cannot equal -3.
So, the restriction is x ≠ -3.
To find the width of the field, we need to divide the area of the field by its length.
The area of the field is given as x^2 + 2x - 15 meters.
The length of the field is given as (x^2 + 3x - 18) / (x + 1) meters.
So, the width (w) of the field can be calculated as:
w = (x^2 + 2x - 15) / ((x^2 + 3x - 18) / (x + 1))
To simplify this, we can multiply by the reciprocal of the denominator:
w = (x^2 + 2x - 15) * (x + 1) / (x^2 + 3x - 18)
Next, we can multiply out the numerator and the denominator:
w = (x^3 + 3x^2 - 13x - 15) / (x^2 + 3x - 18)
Thus, the width of the field is (x^3 + 3x^2 - 13x - 15) / (x^2 + 3x - 18).
Restriction:
To avoid any division by zero, we must make sure that the denominator (x^2 + 3x - 18) is not equal to zero. So, we need to solve the equation:
x^2 + 3x - 18 = 0
By factoring or using the quadratic formula, we find that x = 3 or x = -6. Thus, the restriction is x ≠ -6 and x ≠ 3.
To find the width of the field, we can use the formula for the area of a rectangle, which is A = l × w, where A is the area, l is the length, and w is the width.
In this case, we are given the area of the field as (x^2 + 2x - 15) square meters and the length as (x^2 + 3x - 18) over (x + 1) meters.
Let's set up the equation:
x^2 + 2x - 15 = (x^2 + 3x - 18) / (x + 1)
To simplify the equation, we can first multiply both sides of the equation by (x + 1) to eliminate the denominator:
(x^2 + 2x - 15) × (x + 1) = x^2 + 3x - 18
Expanding both sides of the equation gives:
x^3 + 3x^2 + x^2 + 2x^2 + 2x - 15x - 15 = x^2 + 3x - 18
Combining like terms on both sides:
x^3 + 6x^2 - 13x - 15 = x^2 + 3x - 18
Now, we can gather all terms on one side of the equation:
x^3 + 6x^2 - x^2 - 13x - 3x + 15 + 18 = 0
Simplifying further:
x^3 + 5x^2 - 16x + 33 = 0
Unfortunately, this equation cannot be simplified further and does not factor easily. To find the exact value of x or the width, we would need to use numerical methods or a calculator.
However, there is a restriction in this problem. The denominator (x + 1) in the length should not be equal to zero, as it would result in division by zero. Thus, the restriction is x ≠ -1.
In summary, we have set up the equation and simplified it, but we were unable to find the exact width of the field without using numerical methods or a calculator. The restriction is x ≠ -1.