Question

1. higher order thinking the length of a rectangle is twice the width. the area of the rectangle is 90 square units. note that you can divide the rectangle into two squares.

a. which irrational number represents the length of each side of the squares?
b. estimate the length and width of the rectangle.
image of a blue rectangle with text area = 90 square units

Explanation:

Part a

Step1: Define variables

Let the width of the rectangle be \( w \). Then the length is \( 2w \) (since length is twice the width).

Step2: Use area formula

The area of a rectangle is \( A = \text{length} \times \text{width} \). Given \( A = 90 \), so \( 2w \times w = 90 \).

Step3: Simplify equation

Simplify to \( 2w^2 = 90 \), then divide both sides by 2: \( w^2 = 45 \).

Step4: Solve for \( w \)

Take square root: \( w = \sqrt{45} \) (we take positive root as width is positive). Simplify \( \sqrt{45}=\sqrt{9\times5} = 3\sqrt{5}\approx 6.708 \). Since the rectangle is divided into two squares, the side of each square is equal to the width of the rectangle, so the irrational number representing the side of each square is \( \sqrt{45} \) (or \( 3\sqrt{5} \)).

Part b

Step1: Recall length and width relation

We know length \( l = 2w \) and \( w=\sqrt{45}\approx 6.708 \).

Step2: Calculate length

Length \( l = 2\times\sqrt{45}=2\times3\sqrt{5}=6\sqrt{5}\approx 13.416 \).

Step3: Estimate values

We can also estimate by noting that \( \sqrt{45} \) is between \( \sqrt{36}=6 \) and \( \sqrt{49}=7 \), closer to 7. So width \( \approx 6.7 \), length \( \approx 13.4 \) (since length is twice width).

Part a Answer: The irrational number is \( \boldsymbol{\sqrt{45}} \) (or \( \boldsymbol{3\sqrt{5}} \)) representing the side of each square.

Part b Answer: Width \( \approx \boldsymbol{6.7} \) units, Length \( \approx \boldsymbol{13.4} \) units (or more precisely \( 3\sqrt{5}\approx 6.71 \) and \( 6\sqrt{5}\approx 13.42 \))

Answer:

Step1: Recall length and width relation

We know length \( l = 2w \) and \( w=\sqrt{45}\approx 6.708 \).

Step2: Calculate length

Length \( l = 2\times\sqrt{45}=2\times3\sqrt{5}=6\sqrt{5}\approx 13.416 \).

Step3: Estimate values

We can also estimate by noting that \( \sqrt{45} \) is between \( \sqrt{36}=6 \) and \( \sqrt{49}=7 \), closer to 7. So width \( \approx 6.7 \), length \( \approx 13.4 \) (since length is twice width).

Part a Answer: The irrational number is \( \boldsymbol{\sqrt{45}} \) (or \( \boldsymbol{3\sqrt{5}} \)) representing the side of each square.

Part b Answer: Width \( \approx \boldsymbol{6.7} \) units, Length \( \approx \boldsymbol{13.4} \) units (or more precisely \( 3\sqrt{5}\approx 6.71 \) and \( 6\sqrt{5}\approx 13.42 \))