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Sagot :
Representing a situation as an algebraic equation using variables can be used to solve complex problems
The correct option as option for the depth of the gutter that will allow a cross-sectional area of 54 square inches as option d.
d. y ≈ 2.4 in. and y ≈ 11.1 in.
The reason the selected option is correct is as follows:
The given parameters of the rain gutter are;
The width of the aluminum sheets from which the rain gutter is made = 27 inches
The angle to which the edges are turned up to form right angles = Right angles (90 degrees)
Required:
To determine the depth of the gutter allowing a cross-sectional area of 54 square inches
Method:
Form an equation that describe the area of the gutter, taking note that the gutter does not have a cover (top) and therefore only three sides are required
Solution:
Let y represent the length of the sides (depth) of the rain gutter, and let x represent the width, we have;
2·y + x = 27...(1) (The total with of the gutter; 2 sides and a base)
The cross sectional area of the gutter, A = y × x...(2)
Where;
y = The depth of the gutter
x = The width of the gutter
From equation (1), we have;
x = 27 - 2·y
Therefore, in equation (2), we have;
A = y × (27 - 2·y)
The desired cross-sectional area, A = 54 in.²
∴ A = 54 = y × (27 - 2·y) = 27·y - 2·y²
Which gives;
2·y² - 27·y + 54 = 0
From which we get;
[tex]y = \dfrac{27 \pm \sqrt{(-27)^2 - 4 \times 2 \times 54} }{2 \times 2} = \dfrac{27 \pm 3\times \sqrt{33} }{4}[/tex]
Therefore;
y ≈ 2.4 and/or y ≈ 11.1
Which gives the depth of the gutter that will allow a cross-sectional area of 54 square inches as y ≈ 2.4 in. and y ≈ 11.1 in.
Learn more about algebraic equations here:
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