Question

roderick wants to draw a circle for which the numerical value of the circumference is greater than the value of the area.

which lengths could he use for the radius? select three options.

\\(\frac{1}{2}\\)

x 1

1.5

2

2\\(\frac{1}{2}\\)

Explanation:

Formulate the inequality

We need the circumference to be greater than the area.
Let \(r\) be the radius of the circle.
The circumference is \(2\pi r\).
Using the Area of a Circle concept, the area is \(\pi r^2\).
We set up the inequality:

$$2\pi r > \pi r^2$$

Solve for the radius

Since the radius \(r\) must be positive (\(r > 0\)), we can divide both sides by \(\pi r\):

$$2 > r$$

Thus, the radius must satisfy:

$$0 < r < 2$$

Evaluate the given options

We check which of the given options are strictly less than 2:

• \(\frac{1}{2}\): Since \(0.5 < 2\), this is a valid option.
• \(1\): Since \(1 < 2\), this is a valid option.
• \(1.5\): Since \(1.5 < 2\), this is a valid option.
• \(2\): Since \(2\) is not strictly less than 2, this is invalid.
• \(2\frac{1}{2}\): Since \(2.5 > 2\), this is invalid.

The three correct options are \(\frac{1}{2}\), \(1\), and \(1.5\).

Answer:

• (A) \(\frac{1}{2}\) (Correct answer)
• (B) 1 (Correct answer)
• (C) 1.5 (Correct answer)
• (D) 2
• (E) \(2\frac{1}{2}\)