The perimeter of a quarter circle is 3.57 kilometers. What is the quarter circle’s radius? Use 3.14 for
. P = 3.57 km
The Correct Answer and Explanation is:
To find the radius of a quarter circle with a perimeter of 3.57 kilometers, and using π = 3.14, let’s first understand what the perimeter of a quarter circle includes.
🔵 Formula for the Perimeter of a Quarter Circle:
A quarter circle is one-fourth of a full circle. Its perimeter includes:
- One quarter of the circumference of the circle (curved part),
- And two straight sides (radius lines).
So, the formula is: Perimeter=14(2πr)+2r\text{Perimeter} = \frac{1}{4}(2\pi r) + 2r
Simplifying: Perimeter=12πr+2r\text{Perimeter} = \frac{1}{2} \pi r + 2r
Given:
- Perimeter = 3.57 km
- π = 3.14
Substitute into the formula: 3.57=12⋅3.14⋅r+2r3.57 = \frac{1}{2} \cdot 3.14 \cdot r + 2r 3.57=1.57r+2r3.57 = 1.57r + 2r 3.57=3.57r3.57 = 3.57r
Now, solve for r: r=3.573.57=1r = \frac{3.57}{3.57} = 1
✅ Final Answer:
1 kilometer\boxed{1 \text{ kilometer}}
🔍 Explanation
To determine the radius of a quarter circle when given the perimeter, it’s crucial to understand how the shape is constructed. A quarter circle is a sector of a full circle, cut into four equal parts. Its perimeter includes two parts: the curved arc and the two straight radii that meet at the center.
The arc length is one-fourth of the total circumference of the full circle. The circumference of a circle is calculated using the formula 2πr2\pi r, where rr is the radius and π\pi (pi) is approximately 3.14. Since we’re dealing with a quarter of the circle, the arc length is 14×2πr=12πr\frac{1}{4} \times 2\pi r = \frac{1}{2} \pi r. In addition to the arc, the perimeter includes two straight lines that are both radii of the circle, which add up to 2r2r.
So, the total perimeter of the quarter circle is: 12πr+2r\frac{1}{2} \pi r + 2r
We are told the perimeter is 3.57 kilometers and π is 3.14. Plugging these values into the formula gives: 3.57=1.57r+2r=3.57r3.57 = 1.57r + 2r = 3.57r
Solving for rr, we divide both sides by 3.57, giving us r=1r = 1.
Thus, the radius of the quarter circle is 1 kilometer.
