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.

One-half
1
1.5
2
2 and one-half

Answer 1

To determine which radii will yield a circumference greater than the area of a circle, we can use the following formulas:

- Circumference \(C\) of a circle:
\[
C = 2\pi r
\]

- Area \(A\) of a circle:
\[
A = \pi r^2
\]

We need to find values of \(r\) such that:
\[
C > A \quad \text{or} \quad 2\pi r > \pi r^2
\]

We can simplify this inequality by dividing both sides by \(\pi\) (noting that \(\pi > 0\)):
\[
2r > r^2
\]

Rearranging gives us:
\[
r^2 - 2r < 0
\]

Factoring:
\[
r(r - 2) < 0
\]

This inequality holds true when \(0 < r < 2\).

Now we can evaluate each of the proposed radii:

1. **One-half \( (0.5) \)**: This is in the range \(0 < r < 2\).
2. **1**: This is also in the range \(0 < r < 2\).
3. **1.5**: This is in the range \(0 < r < 2\).
4. **2**: This is \(r = 2\), which does not satisfy \(r < 2\).
5. **Two and one-half \( (2.5) \)**: This is above the upper limit of \(r < 2\).

Thus, the radii Roderick can choose for which the circumference is greater than the area are:

- One-half
- 1
- 1.5

So the three options are: **One-half, 1, and 1.5**.