Respuesta :
Answer:
[tex]Length=17.5\ in\\\\Width=11.5\ in[/tex]
Step-by-step explanation:
The formula for calculate the Area of a rectangle is:
[tex]A=lw[/tex]
Where "l" is the lenght and "w" is the width.
And the formula for calculate the peimeter of a rectangle is:
[tex]P=2l+2w[/tex]
Where "l" is the lenght and "w" is the width.
We know that the perimeter of the rectangular baking sheet is 58 inches and its area is 201.25 in². Then:
[tex]A=201.25\ in^2\\\\P=58\ in[/tex]
The steps are:
1. Solve for the "l" from the formula [tex]A=lw[/tex]:
[tex]A=lw\\\\l=\frac{A}{w}\\\\\l=\frac{201.25}{w}[/tex]
2. Substitute [tex]l=\frac{201.25}{w}[/tex] into the formula [tex]P=2l+2w[/tex] and solve for "w":
[tex]58=2(\frac{201.25}{w})+2w\\\\58=\frac{402.5}{w}+2w\\\\58=\frac{402.5+2w^2}{w}\\\\58w=402.5+2w^2\\\\2w^2-58w+402.5=0[/tex]
Applying the Quadratic formula [tex]x=\frac{-b\±\sqrt{b^2-4ac}}{2a}[/tex], we get:
[tex]x=w=\frac{-(-58)\±\sqrt{(-58)^2-4(2)(402.5)}}{2(2)}\\\\w_1=17.5\\\\w_2=11.5[/tex]
3. Substitute [tex]w_1=17.5[/tex] into [tex]l=\frac{201.25}{w}[/tex]:
[tex]l=\frac{201.25}{17.5}=11.5[/tex]
4. Substitute [tex]w_2=11.5[/tex] into [tex]l=\frac{201.25}{w}[/tex]:
[tex]l=\frac{201.25}{11.5}=17.5[/tex]
Therefore, since the value of the lenght of a rectangle must be greater that the value of the width, we can conclude that the lenght and the width of the rectangular baking sheet are:
[tex]l=17.5\ in\\\\w=11.5\ in[/tex]
The length and width of the baking sheet are 17.5 inches and 11.5 inches respectively.
The perimeter of a rectangle is given as: 2l + 2w
The area of a rectangle is given as l×w
2l + 2w = 58
l + w = 29 ....... equation i
l × w = 201.25 ....... equation ii
From equation i, l = 29 - w
Since l×w = 201.25
(29 - w)w = 201.25
w² - 29w = 201.25
w² - 29w - 201.25 = 0
w² - 17.5w - 11.5w - 201.25 = 0
w(w - 17.5) - 11.5(w - 17.5)
(w - 17.5) = 0 and (w - 11.5) = 0
w - 11.5 = 0
w = 0 + 11.5 = 11.5
Width = 11.5 inches
Length = 17.5 inches
Therefore, the length and width of the baking sheet are 17.5 inches and 11.5 inches respectively.
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