Respuesta :
Answer:
Length of rectangle = 24 inches
Width of rectangle = 1 inches
Step-by-step explanation:
Area of rectangle = 24 square inches
Perimeter of rectangle = 50 inches
Formulas:
Area of rectangle = [tex]l\times w[/tex]
Perimeter of rectangle = [tex]2(l+w)[/tex]
where [tex]l[/tex] represents length of rectangle and [tex]w[/tex] represents width of rectangle.
So, we can get two equations for [tex]l[/tex] and [tex]w[/tex]
A) [tex]l\times w=24[/tex] [Area]
B) [tex]2(l+w)=50[/tex] [ Perimeter ]
Simplifying equation B.
Dividing both sides by 2.
[tex]\frac{2(l+w)}{2}=\frac{50}{2}[/tex]
[tex]l+w=25[/tex]
Solving for [tex]l[/tex] in terms of width.
Subtracting [tex]w[/tex] both sides.
[tex]l+w-w=25-w[/tex]
∴ [tex]l=25-w[/tex]
Substituting value of [tex]l[/tex] in terms of [tex]w[/tex] in equation A.
[tex](25-w)w=24[/tex]
Using distribution.
[tex]25w-w^2=24[/tex]
Adding [tex]w^2[/tex] both sides.
[tex]25w-w^2+w^2=24+w^2[/tex]
[tex]25w=24+w^2[/tex]
subtracting [tex]25 w[/tex] both sides
[tex]25w-25w=24+w^2-25w [/tex]
[tex]0=w^2-25w+24[/tex]
We have a quadratic equation. [tex]w^2-25w+24=0[/tex]
We solve for [tex]w[/tex] by factor method.
Splitting middle term into two terms such that the sum of the two =[tex]-25w[/tex] and product of two = [tex]24w^2[/tex]
[tex]w^2-24w-w+24=0[/tex]
Factoring in pairs.
[tex]w(w-24)-1(w-24)=0[/tex]
Factoring as a whole
[tex](w-24)(w-1)=0[/tex]
So, we have
[tex]w-24=0[/tex] and [tex]w-1=0[/tex]
By adding 24 to one equation and adding 1 to other we get
[tex]w-24+24=0+24[/tex] and [tex]w-1+1=0+1[/tex]
[tex]w=24[/tex] and [tex]w=1[/tex]
So length can be found out by plugging value of [tex]w[/tex] in the length equation.
[tex]l=25-24=1[/tex] and [tex]l=25-1=24[/tex]
Since length is always the longer side of rectangle so, the dimensions of rectangle can be given by.
length = 24 inches (answer)
width = 1 inches (answer)
