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Answer :
The dimensions of the workbook are [tex]6[/tex] inches and [tex]8[/tex] inches.
To find the dimensions of the workbook given the area and perimeter, we can use the following formulas:
Area: The area of a rectangle is calculated using the formula:
[tex]A = l \times w[/tex]
where [tex]l[/tex] is the length and [tex]w[/tex] is the width.
Perimeter: The perimeter of a rectangle is calculated using the formula:
[tex]P = 2(l + w)[/tex]
where [tex]l[/tex] is the length and [tex]w[/tex] is the width.
Given:
Area [tex]A = 48[/tex] square inches
Perimeter [tex]P = 28[/tex] inches
Step 1: Set up the equations.
From the area equation:
[tex]l \times w = 48 \quad (1)[/tex]
From the perimeter equation:
[tex]2(l + w) = 28 \quad (2)[/tex]
Dividing this equation by 2 gives:
[tex]l + w = 14 \quad (3)[/tex]
Step 2: Solve the system of equations.
From equation (3), we can express [tex]l[/tex] in terms of [tex]w[/tex]:
[tex]l = 14 - w \quad (4)[/tex]
Step 3: Substitute equation (4) into equation (1).
Substituting [tex]l[/tex] from equation (4) into equation (1):
[tex](14 - w)w = 48[/tex]
Rearranging gives:
[tex]14w - w^2 = 48[/tex]
or
[tex]w^2 - 14w + 48 = 0[/tex]
Step 4: Solve the quadratic equation.
To factor this quadratic equation, we need two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8. Thus, we can factor it as:
[tex](w - 6)(w - 8) = 0[/tex]
Setting each factor equal to zero gives us the possible widths:
[tex]w = 6 \quad or \quad w = 8[/tex]
Step 5: Find the corresponding lengths.
Using equation (4), if [tex]w = 6[/tex], then:
[tex]l = 14 - 6 = 8[/tex]
And if [tex]w = 8[/tex], then:
[tex]l = 14 - 8 = 6[/tex]
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