MEGA Elementary Education Multi-Content Practice Test

Which expression represents the area of the smallest sector formed by the minute and hour hand on an 8-inch clock face when it reads 9:00?

1. (1/4)𝝅4^2

2. (1/4)𝝅8^2

3. (1/2)𝝅4^2

4. (1/2)𝝅8^2

The correct answer is: (1/4)𝝅4^2

To understand why the correct expression represents the area of the smallest sector formed by the minute and hour hands at 9:00 on an 8-inch clock face, we must first analyze the position of the hands. At 9:00, the hour hand points at the 9, and the minute hand points at the 12. This forms a right angle (90 degrees) between the two hands. The geometry of a sector can be described using the formula for the area of a circle, which is \(A = \pi r^2\), where \(r\) is the radius. Since the clock has an 8-inch radius, the area of the entire circle would be \(A = \pi (8^2) = 64\pi\). However, we are interested in the area of the sector formed by the hour and minute hand at a right angle. A full circle is 360 degrees, and since the sector at 9:00 represents 90 degrees, it is a quarter of the full circle. Therefore, the area of the sector can be calculated as: \[ \text{Area of sector} = \frac{90}{360} \times \text{Total area} =