Final answer:
The area of the sector formed by the hands of a clock at 5:00 can be calculated by finding the angle between the hands (which is 150 degrees for 5:00) and using it in the sector area formula. The total area of the clock is given to solve for the radius, and the sector area formula is then used to find the area of the sector, rounding to the nearest tenth.
Step-by-step explanation:
To calculate the area of the sector formed by the hands of the clock at 5:00, we first need to determine the angle between the hour hand and the minute hand at that time. At 5:00, the minute hand is at the 12 and the hour hand is at the 5. Since each hour represents 30 degrees (360 degrees divided by 12 hours), the angle between the hands is 5 hours × 30 degrees/hour, which is 150 degrees.
Knowing that the area of the full circle is 452.4 square inches, we can use the formula for the area of a sector (sector area) of a circle: sector area = (θ/360) × (πr^2), where θ is the angle in degrees and r is the radius of the circle. To find the radius (r) of the clock, we use the area formula for the circle, A = πr^2, to solve for r.
Let's find the radius first: 452.4 = πr^2. Solving for r gives us the radius of the clock. Once we have the radius, we substitute it and θ = 150 into the sector area formula to compute the area of the sector, which corresponds to the area 'swept' by the clock hands at 5:00. Finally, we round the result to the nearest tenth to get our answer.