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.
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 the length of rectangle b?

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle is given by the formula \( A = \text{length} \times \text{width} \). For Rectangle A, the length is \( (2x + 7) \) cm and the width is \( 4x \) cm.

Step2: Calculate area of Rectangle A

Multiply the length and width: \( (2x + 7) \times 4x \). Using the distributive property (also known as the distributive law of multiplication over addition), we get \( 2x \times 4x + 7 \times 4x = 8x^2 + 28x \). So the area of Rectangle A is \( 8x^2 + 28x \) square centimeters.

Step3: Find area of Rectangle B (part b)

We know that the area of Rectangle B is three times the area of Rectangle A. So we multiply the area of Rectangle A by 3. That is \( 3 \times (8x^2 + 28x) \). Using the distributive property again, we get \( 3 \times 8x^2 + 3 \times 28x = 24x^2 + 84x \). So the area of Rectangle B is \( 24x^2 + 84x \) square centimeters.

Step4: Find length of Rectangle B (part c)

The area of a rectangle is also \( A=\text{width} \times \text{length} \), so \( \text{length}=\frac{\text{area}}{\text{width}} \). The width of Rectangle B is \( 12x \) and the area of Rectangle B is \( 24x^2 + 84x \). First, factor out the greatest common factor from the area: \( 24x^2 + 84x = 12x(2x + 7) \). Then divide this by the width \( 12x \): \( \frac{12x(2x + 7)}{12x} \). The \( 12x \) terms cancel out (assuming \( x
eq0 \), which is valid in the context of a length/width being non - zero), leaving \( 2x + 7 \). So the length of Rectangle B is \( 2x + 7 \) centimeters.

Answer:

s:
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.