The figure below is made up of 3 similar quadrants. Its perimeter is 61. Find its radius.  (Take )

1. Break Down the Problem

The problem states that we have three similar quadrants, and we need to find the radius given that the total perimeter is 61. We will need to:

1. Understand what a quadrant is and how it relates to the circle.
2. Determine the perimeter contribution from each quadrant.
3. Set up an equation based on the total perimeter.

2. Relevant Concepts

A quadrant is a quarter of a circle. The perimeter $P$ of one quadrant consists of the curved part (which is $\frac{1}{4}$ of the circumference of the full circle) plus the two straight edges (both equal to the radius $r$).

The circumference $C$ of a full circle is given by: $C = 2\pi r$ Thus, the perimeter of one quadrant would be: $P_{\text{quadrant}} = \frac{1}{4}(2\pi r) + 2r = \frac{\pi r}{2} + 2r$

3. Analysis and Detail

Since there are three similar quadrants, the total perimeter can be expressed as: $P_{\text{total}} = 3 \left( \frac{\pi r}{2} + 2r \right)$

We know from the problem that: $P_{\text{total}} = 61$

Setting the equation up: $3 \left( \frac{\pi r}{2} + 2r \right) = 61$

4. Verify and Summarize

Now we can simplify and solve for $r$:

1. Distribute the 3: $\frac{3\pi r}{2} + 6r = 61$

2. To eliminate the fraction, multiply through by 2: $3\pi r + 12r = 122$

3. Combine like terms: $(3\pi + 12) r = 122$

4. Now solve for $r$: $r = \frac{122}{3\pi + 12}$

Plugging the approximate value of $\pi \approx 3.14$: $r \approx \frac{122}{3(3.14) + 12}$ Calculating: $r \approx \frac{122}{9.42 + 12} = \frac{122}{21.42} \approx 5.69$

Final Answer

The radius $r$ is approximately $5.69$.