Question

$$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ \text{The circumference of a circular region with radius r} & \text{The perimeter of a square with side r}\end{array}$$ a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

Short answer

The answer is (b). Quantity B is greater than Quantity A.

Step-by-step solution

Write down the formulas

Start by writing down the formulas for both quantities. The formula for the circumference of a circle is \(C = 2\pi r\), and the formula for the perimeter of a square is \(P = 4r\). Here, \(r\) is the radius of the circle and the side of the square.

Compare the constants

Before comparing the quantities, compare the constants in the formulas, i.e., \(\pi\) in the circle's circumference formula and \(4\) in the square's perimeter formula. Because \(\pi = 3.1416\) and \(4 > \pi\), it's clear that the square's perimeter would be larger if \(r\) is the same for both.

Decision

Since the square's perimeter outnumbers the circle's circumference due to the larger constant, Quantity B is greater than Quantity A. Therefore, the answer is choice b.