Final answer:
To calculate the sector area at 4:00, find the angle between the clock's hands (120 degrees or 2\(\pi\)/3 radians), then use the sector area formula with a radius of 9 inches, resulting in an area of approximately 85.95 square inches.
Step-by-step explanation:
To calculate the sector area created by the hands of a clock at 4:00, you need to find the angle between the hour and minute hand and then use the formula Area = (\(\theta\) in radians/2) \(\times\) radius^2. At 4:00, the minute hand is at the 12 and the hour hand is at the 4, which is 1/3 of the way between 12 and 6. This creates a 120-degree angle (4 hours at 30 degrees per hour). Since we need the angle in radians for the formula, we convert degrees to radians by multiplying by \(\pi\)/180, resulting in 120 degrees \(\times\) \(\pi\)/180 = 2\(\pi\)/3 radians. Then we can calculate the sector area using the given radius of the clock (9 inches).
Thus, the sector area A is given by:
A = (2\(\pi\)/3 radians / 2) \(\times\) (9 inches)^2 = (\(\pi\)/3) \(\times\) 81 inches^2 = 81\(\pi\)/3 inches^2 \approx 85.95 inches^2.
This calculation shows the importance of understanding angles and their conversions, as well as the use of formulas to determine areas in practical applications like reading a clock.