Question

1. a squares side length is equal to a circles radius. molly graphed the relationship between the squares side length and its perimeter and the relationship between a circles radius and its circumference. which line is steeper? would your response change if the squares side length was equal to the circles diameter?

Explanation:

Step1: Find the equations

Let the side length of the square (and radius of the circle) be \(x\).

• Perimeter of square \(P = 4x\), slope \(m_1=4\).
• Circumference of circle \(C = 2\pi x\approx6.28x\), slope \(m_2 = 2\pi\approx6.28\).

Step2: Compare slopes

Since \(6.28>4\), the line for the circle's circumference is steeper.

If the square's side length \(s\) is equal to the circle's diameter \(d\) (so radius \(r=\frac{s}{2}\)):

• Perimeter of square \(P = 4s\), slope \(m_1 = 4\).
• Circumference of circle \(C=2\pi r=\pi s\approx3.14s\), slope \(m_2=\pi\approx3.14\).

Since \(4 > 3.14\), the line for the square's perimeter is steeper.

Answer:

Initially, the line for the circle's circumference is steeper. When the square's side length is equal to the circle's diameter, the line for the square's perimeter is steeper.