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 (2)
Determine the angle between the hour and minute hands at 4:00, which is 120 degrees.
Convert the angle 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 radians.
Calculate the area: A = 2 1 ( 9 2 ) ( 3 2 π ) = 27 π square inches. The final answer is 27 πi n 2 .
Explanation
Problem Analysis Let's analyze the problem. We need to find the area of the sector formed by the hands of a clock at 4:00. The radius of the clock is 9 inches. The minute hand is at 12 and the hour hand is at 4.
Calculate the angle First, we need to find the angle between the hour and minute hands. Since there are 12 numbers on the clock, each number is separated by an angle of 12 360 = 30 degrees. At 4:00, the angle between the hands is 4 × 30 = 120 degrees.
Convert to radians Now, we need to convert the angle from degrees to radians. To do this, we multiply the angle in degrees by 180 π . So, 120 × 180 π = 3 2 π radians.
Apply the sector area formula Next, we use the formula for the area of a sector, which is A = 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.
Calculate the area Substitute the values into the formula: A = 2 1 ( 9 2 ) ( 3 2 π ) = 2 1 ( 81 ) ( 3 2 π ) = 3 81 π = 27 π square inches.
Final Answer Therefore, the area of the sector formed by the hands of the clock at 4:00 is 27 π square inches.
Examples
Imagine you're designing a pizza cut into 12 slices, like the hours on a clock. If you eat 4 slices (representing 4 'hours'), you've eaten a sector of the pizza. Knowing how to calculate the area of this sector helps you determine exactly how much pizza you consumed, based on the pizza's radius and the angle of the slices you ate. This is a tasty application of sector area calculations!
The area of the sector formed by the clock hands at 4:00 is 27 π square inches . This corresponds to option C. To calculate this, we determined the angle between the hands, converted it to radians, and applied the sector area formula.
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