Final answer:
To find the area of the sector formed by the clock's hands at 5:00, calculate the angle of 150 degrees between the hands, determine the radius of the clock using the given area, and then apply the formula for the area of a sector.
Step-by-step explanation:
To determine the area of the sector formed by the hands of the clock when it is 5:00, we first need to calculate the angle between the hour and minute hands at that time. At 5:00, the angle between the hour and minute hands is 150 degrees because the hour hand is 5/12 of the way through its cycle (5 hours out of 12), corresponding to 5/12 of 360 degrees, which is 150 degrees. The area of the sector formed can be found using the formula for the area of a sector: Area of a sector = (θ/360) × π radius2, where θ is the angle in degrees. First, we must find the radius of the clock.
Given the area of the clock face is 452.4 in2, we can use the area of a circle, A = πr2, to find the radius. Substituting the given area and solving for r:
r = (√(A/π))
Once we have the radius, we can use it to find the area of the sector using our formula:
Area of the sector = (150/360) × π × r2
Calculating this will give us the area of the sector at 5:00.