The perimeter of a quarter circle is 3.57 kilometers. What is the quarter circle’s radius? Use 3.14 for . kilometers Siubmit explain
The Correct Answer and Explanation is :
To find the radius of a quarter circle when the perimeter is given, we first need to understand the components that make up the perimeter of a quarter circle. The perimeter (P) of a quarter circle includes two straight sides (the radius) and the curved part (the arc).
- Understanding the Perimeter:
The formula for the perimeter (P) of a quarter circle can be expressed as:
[
P = r + r + \frac{1}{4}(2\pi r)
]
Here:
- (r) is the radius,
- The two (r) terms account for the two straight sides (radii),
- The term (\frac{1}{4}(2\pi r)) represents the arc length of the quarter circle. Simplifying the arc length term:
[
\frac{1}{4}(2\pi r) = \frac{\pi r}{2}
]
Thus, the complete formula for the perimeter becomes:
[
P = 2r + \frac{\pi r}{2}
]
- Substituting the Given Perimeter:
We know that the perimeter (P) is 3.57 kilometers, and we can substitute this value into the equation:
[
3.57 = 2r + \frac{3.14r}{2}
] - Solving for (r):
To eliminate the fraction, multiply the entire equation by 2:
[
2 \times 3.57 = 4r + 3.14r
]
This simplifies to:
[
7.14 = 4r + 3.14r
]
Combining like terms, we get:
[
7.14 = 7.14r
]
Dividing both sides by 7.14 gives:
[
r = 1
] - Conclusion:
Therefore, the radius of the quarter circle is 1 kilometer.
This result shows that a quarter circle with a total perimeter of 3.57 kilometers has a radius of 1 kilometer. Understanding the formula and how to manipulate it is crucial when working with geometric shapes and their properties.