Question

1. the perimeter of the rectangle is represented by 8y metres and the area is represented by (6y + 3) square metres. a) write two equations in terms of x and y: one for the perimeter and one for the area of the rectangle. b) determine the perimeter and the area.

Explanation:

Step1: Write the perimeter formula

The perimeter \( P \) of a rectangle is \( P = 2(l + w) \), where \( l=x + 8 \) and \( w=x + 6 \).

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Step2: Write the area formula

The area \( A \) of a rectangle is \( A=l\times w \), where \( l=x + 8 \) and \( w=x + 6 \).

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Step3: Solve the system of equations

From \( 8y=4x + 28 \), we get \( y=\frac{4x + 28}{8}=\frac{x + 7}{2} \).
Substitute \( y=\frac{x + 7}{2} \) into \( 6y+3=x^{2}+14x + 48 \):

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We take \( x=-3 \) (since length cannot be negative in a real - world context for these expressions).
Substitute \( x = - 3 \) into \( y=\frac{x + 7}{2} \), \( y=\frac{-3 + 7}{2}=2 \).

Step4: Calculate the perimeter and area

For the perimeter: \( P = 8y \), substitute \( y = 2 \), \( P=16 \) metres.
For the area: \( A=6y+3 \), substitute \( y = 2 \), \( A=6\times2+3=15 \) square metres.

Answer:

a) Perimeter equation: \(8y = 4x + 28\), Area equation: \(6y+3=x^{2}+14x + 48\)
b) Perimeter: \(16\) metres, Area: \(15\) square metres