I have 2 questions. one is a rectangle that has an area of 34 square meters, the width is (x+4)m and the length is (x+11)m. The other problem is a triangle that has an area of 119 square ft. The legs are (x+3) ft and (x+6) ft. It has a right angle opposite the hippotenuse. They both ask to find the dimensions. This is in the factoring trinomials using xsquared + bx + c
(x+4)(x+11) = 34
x^2 + 15 x + 44 = 34
x^2 + 15 x + 10 = 0
x = [ -15 + /- sqrt (225 -40)/2
x = [ -15 +/- 13.6 ] / 2
x = -14.3 or - .7
discard -14.3 because negative length and with would result
so use x = -.7
(1/2)(x+3)(x+6) = 119
x^2 + 9 x + 18 = 238
x^2 + 9 x - 220 = 0
(x+20)(x-11) = 0
x = 11
To solve both problems using factoring trinomials, we need to find the value of 'x' and then use it to calculate the dimensions.
1. Solving for the rectangle's dimensions:
Given that the area of the rectangle is 34 square meters, and the width is (x+4) meters while the length is (x+11) meters, we can set up the equation:
Area = Length × Width
34 = (x+11)(x+4)
To find the value of 'x', we need to factor the quadratic equation, set it equal to zero, and solve for 'x'.
First, expand the equation:
34 = x^2 + 15x + 44
Rearrange to set the equation equal to zero:
x^2 + 15x + 44 - 34 = 0
x^2 + 15x + 10 = 0
Now, factor the quadratic equation:
(x + 5)(x + 2) = 0
Set each factor equal to zero and solve for 'x':
x + 5 = 0 or x + 2 = 0
x = -5 or x = -2
Since dimensions cannot be negative, we discard the negative values for 'x'.
For 'x = -5', the dimensions would be:
Width = x + 4 = -5 + 4 = -1, which is not valid.
Length = x + 11 = -5 + 11 = 6
For 'x = -2', the dimensions would be:
Width = x + 4 = -2 + 4 = 2
Length = x + 11 = -2 + 11 = 9
The dimensions of the rectangle would be width = 2 meters and length = 9 meters.
2. Solving for the triangle's dimensions:
Given that the area of the triangle is 119 square feet, and the legs are (x+3) feet and (x+6) feet that form a right angle, we can set up the equation:
Area = 1/2 × Base × Height
119 = 1/2 × (x+3) × (x+6)
Expand the equation:
238 = (x+3)(x+6)
Set the equation equal to zero:
x^2 + 9x + 18 = 0
Factor the quadratic equation:
(x + 6)(x + 3) = 0
Set each factor equal to zero and solve for 'x':
x + 6 = 0 or x + 3 = 0
x = -6 or x = -3
Since dimensions cannot be negative, we discard the negative values for 'x'.
For 'x = -6', the dimensions would be:
Base = x + 3 = -6 + 3 = -3, which is not valid.
Height = x + 6 = -6 + 6 = 0, which is not valid.
For 'x = -3', the dimensions would be:
Base = x + 3 = -3 + 3 = 0, which is not valid.
Height = x + 6 = -3 + 6 = 3
The dimensions of the triangle would be base = 0 feet and height = 3 feet.
In summary:
For the rectangle, the dimensions are width = 2 meters and length = 9 meters.
For the triangle, the dimensions are base = 0 feet and height = 3 feet.