Question
A circular spinner has a radius of 6 inches. The spinner is divided into three sections of unequal area. The sector labeled green has a central angle of 90°. A point on the spinner is randomly selected.
What is the probability that the randomly selected point falls in the green sector?
What is the probability that the randomly selected point falls in the green sector?
Answers
GPT-4o mini
To find the probability that a randomly selected point falls in the green sector of the circular spinner, we first need to determine the area of the entire spinner and the area of the green sector.
1. **Calculate the area of the entire spinner:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius. The radius of the spinner is given as 6 inches.
\[
A = \pi (6)^2 = 36\pi \text{ square inches}
\]
2. **Calculate the area of the green sector:**
The area of a sector is given by the formula:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta \) is the central angle of the sector in degrees. For the green sector, \( \theta = 90^\circ \).
Substituting the values:
\[
\text{Area of green sector} = \frac{90}{360} \times \pi (6)^2
\]
Simplifying:
\[
\text{Area of green sector} = \frac{1}{4} \times 36\pi = 9\pi \text{ square inches}
\]
3. **Calculate the probability:**
The probability \( P \) of the randomly selected point falling within the green sector is the ratio of the area of the green sector to the area of the entire spinner:
\[
P = \frac{\text{Area of green sector}}{\text{Area of entire spinner}} = \frac{9\pi}{36\pi}
\]
The \( \pi \) in both the numerator and denominator cancels out:
\[
P = \frac{9}{36} = \frac{1}{4}
\]
Thus, the probability that the randomly selected point falls in the green sector is \( \boxed{\frac{1}{4}} \).
1. **Calculate the area of the entire spinner:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius. The radius of the spinner is given as 6 inches.
\[
A = \pi (6)^2 = 36\pi \text{ square inches}
\]
2. **Calculate the area of the green sector:**
The area of a sector is given by the formula:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta \) is the central angle of the sector in degrees. For the green sector, \( \theta = 90^\circ \).
Substituting the values:
\[
\text{Area of green sector} = \frac{90}{360} \times \pi (6)^2
\]
Simplifying:
\[
\text{Area of green sector} = \frac{1}{4} \times 36\pi = 9\pi \text{ square inches}
\]
3. **Calculate the probability:**
The probability \( P \) of the randomly selected point falling within the green sector is the ratio of the area of the green sector to the area of the entire spinner:
\[
P = \frac{\text{Area of green sector}}{\text{Area of entire spinner}} = \frac{9\pi}{36\pi}
\]
The \( \pi \) in both the numerator and denominator cancels out:
\[
P = \frac{9}{36} = \frac{1}{4}
\]
Thus, the probability that the randomly selected point falls in the green sector is \( \boxed{\frac{1}{4}} \).