PAC-MAN is a circle of radius 24 with a sector of angle π/3 removed. PAC-MAN eats circular pellets of radius 2. If PAC-MAN wants to eat pellets that have that same total area that he does, how many pellets must he eat?
area of PAC-MAN = (5/18) (π(24)^2)
=160π
area of 1 pellet = 4π
number of pellets equal to PAC-MAN = 160π/(4π) = 40
The area of a circle of radius 24 is 576π. The missing sector is 16 of the circle, so the area of PAC-MAN is 480π. A circular pellet of radius 2 has area 4π, so the number that PAC-MAN needs to eat is 480π4π=120.
To determine the number of pellets PAC-MAN must eat in order to have the same total area as himself, we need to break down the problem step by step.
Step 1: Calculate the area of PAC-MAN's missing sector
The area of a sector can be calculated using the formula: A = (θ/2) * r^2, where θ is the angle in radians and r is the radius.
In this case, the angle (θ) of the missing sector is π/3, and the radius (r) is 24.
So, the area of the missing sector is A = (π/3) * 24^2.
Step 2: Calculate the total area of PAC-MAN
The total area of PAC-MAN is equal to the area of the entire circle minus the area of the missing sector.
The area of a circle can be calculated using the formula: A = π * r^2, where π is approximately 3.14159 and r is the radius.
So, the total area of PAC-MAN is A = π * 24^2 - (π/3) * 24^2.
Step 3: Calculate the area of a single pellet
The area of a circular pellet can be calculated using the formula: A = π * r^2, where π is approximately 3.14159 and r is the radius.
In this case, the radius of the pellet is 2.
So, the area of a single pellet is A = π * 2^2.
Step 4: Determine the number of pellets PAC-MAN must eat
To find the number of pellets PAC-MAN must eat, we divide the total area of PAC-MAN by the area of a single pellet and round up to the nearest whole number.
Number of pellets = total area of PAC-MAN / area of a single pellet.
Let's go ahead and calculate the answer:
- Calculate the area of the missing sector:
A1 = (π/3) * 24^2
- Calculate the total area of PAC-MAN:
A2 = π * 24^2 - (π/3) * 24^2
- Calculate the area of a single pellet:
A3 = π * 2^2
- Determine the number of pellets PAC-MAN must eat:
Number of pellets = A2 / A3
Now, let's plug in the values and calculate the answer.