What is the sector area created by the hands of a clock with a radius of 9 inches when the time is $4:00$?
A. $6.75 \pi in ^2$
B. $20.25 \pi in ^2$
C. $27 \pi in ^2$
D. $81 \pi in ^2$
Answer (1)
Determine the angle between the hour and minute hands at 4:00, which is 120 degrees.
Convert the angle from degrees to radians: 12 0 ∘ = 3 2 π radians.
Apply the formula for the area of a sector: A = 2 1 r 2 θ , where r = 9 inches and θ = 3 2 π .
Calculate the sector area: A = 2 1 ( 9 2 ) ( 3 2 π ) = 27 π square inches. The final answer is 27 πi n 2 .
Explanation
Problem Analysis We are given a clock with a radius of 9 inches, and we need to find the area of the sector formed by the hands of the clock at 4:00.
Calculate the Angle At 4:00, the minute hand is at 12 and the hour hand is at 4. There are 12 numbers on a clock, so each number represents 12 360 = 30 degrees. The angle between the hands is therefore 4 × 30 = 120 degrees.
Convert to Radians To use the formula for the area of a sector, we need the angle in radians. To convert from degrees to radians, we use the formula: radians = degrees × 180 π . So, 120 degrees = 120 × 180 π = 3 2 π radians.
Calculate the Area Now we can calculate the area of the sector using the formula: Area = 2 1 r 2 θ , where r is the radius and θ is the angle in radians. In this case, r = 9 inches and θ = 3 2 π radians. So, Area = 2 1 × 9 2 × 3 2 π = 2 1 × 81 × 3 2 π = 27 π square inches.
Final Answer Therefore, the sector area created by the hands of the clock at 4:00 is 27 π square inches.
Examples
Understanding sector areas is useful in many real-world scenarios. For example, if you're designing a sprinkler system for a circular lawn, you need to calculate the area each sprinkler head will cover to ensure the entire lawn is watered efficiently. Similarly, chefs use sector area calculations when slicing pies or pizzas to ensure each slice is the same size. These calculations help in resource allocation and fairness in everyday tasks.
