Answer:
Step-by-step explanation:
Let x = width in ft
y= length in ft
Given, y = 3+2x ----(1)
Area = xy = 77 ft2
Using (1) to replace y
3x+2x2=77
2x2+3x-77=0
Solve this quadratic equation, a=2, b=3, c=-77
x=(-3(+or-)sqrt(32-4*2*(-77)))/(2*2)
= (-3 (+ or -) sqrt(9+616))/4
= (-3 (+ or -) sqrt(625))/4
= (-3 (+ or -) 25)/4
x = (-3+25)/4 or x = (-3-25)/4
= 22/4 = -28/4
x = 5.5 x = -7
x is the length and can't be negative. so, proceed with x= 5.5
xy = 77
y = 77/5.5 = 14
Dimensions of the rectangle are 5.5ft and 14 ft
(Check: 2* 5.5 + 3 = 11+3 = 14)
The area of a shape is the amount of space it occupies.
The dimension of the rectangle is: 14m by 5.5m
The given parameters are:
[tex]\mathbf{Area = 77m^2}[/tex]
[tex]\mathbf{Length = 3 + 2Width}[/tex]
So, the area of the rectangle is:
[tex]\mathbf{Area = Length \times Width}[/tex]
Substitute [tex]\mathbf{Length = 3 + 2Width}[/tex]
[tex]\mathbf{Area = (3 + 2Width) \times Width}[/tex]
Substitute [tex]\mathbf{Area = 77m^2}[/tex]
[tex]\mathbf{77 = (3 + 2Width) \times Width}[/tex]
Replace Width with w
[tex]\mathbf{77 = (3 + 2w) \times w}[/tex]
Open brackets
[tex]\mathbf{77 = 3w + 2w^2}[/tex]
Rewrite as:
[tex]\mathbf{2w^2 + 3w - 77 = 0}[/tex]
[tex]\mathbf{2w^2 + 14w - 11w - 77 = 0}[/tex]
Factorize
[tex]\mathbf{2w(w + 7) - 11(w + 7) = 0}[/tex]
Factor out w + 7
[tex]\mathbf{(2w - 11) (w + 7) = 0}[/tex]
Split
[tex]\mathbf{(2w - 11) = 0\ or\ (w + 7) = 0}[/tex]
Solve for w
[tex]\mathbf{2w = 11\ or\ w = -7}[/tex]
Width cannot be negative.
So, we have:
[tex]\mathbf{2w = 11}[/tex]
Divide through by 2
[tex]\mathbf{w = 5.5}[/tex]
Substitute 5.5 for width in [tex]\mathbf{Length = 3 + 2Width}[/tex]
[tex]\mathbf{Length = 3 + 5.5 \times 2}[/tex]
[tex]\mathbf{Length = 14}[/tex]
Hence, the dimension of the rectangle is: 14m by 5.5m
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