Question

1. the area of rectangle b is three times the area of rectangle a. provide each of your answers in simplified form.

(2x+7)cm rectangle a
(4x) cm rectangle b
a) write an expression to represent the area of rectangle a.
4x(2x+7) = 8x² + 28x
b) determine an expression for the area of rectangle b.
c) if the width of rectangle b can be represented by 12x, what is an expression for the length of rectangle b?

Explanation:

Step1: Find area of Rectangle A

The area of a rectangle is length times width. For Rectangle A, length is \((4x)\) cm and width is \((2x + 7)\) cm. So, area \(A=\text{length}\times\text{width}=4x\times(2x + 7)\). Using distributive property (\(a(b + c)=ab+ac\)), we get \(4x\times2x+4x\times7 = 8x^{2}+28x\).

Step2: Find area of Rectangle B

Given that area of Rectangle B is three times the area of Rectangle A. So, area of B \(=3\times(8x^{2}+28x)\). Distribute the 3: \(3\times8x^{2}+3\times28x=24x^{2}+84x\).

Step3: Find length of Rectangle B

The area of a rectangle is also length times width. For Rectangle B, area is \(24x^{2}+84x\) and width is \(12x\). Let length be \(l\). So, \(l\times12x = 24x^{2}+84x\). Divide both sides by \(12x\): \(l=\frac{24x^{2}+84x}{12x}\). Split the fraction: \(\frac{24x^{2}}{12x}+\frac{84x}{12x}\). Simplify each term: \(2x + 7\).

Answer:

a) The expression for the area of Rectangle A is \(\boldsymbol{8x^{2}+28x}\) square centimeters.
b) The expression for the area of Rectangle B is \(\boldsymbol{24x^{2}+84x}\) square centimeters.
c) The expression for the length of Rectangle B is \(\boldsymbol{2x + 7}\) centimeters.