A circular coaster, with a diameter of 7 inches, is positioned on a white placemat. The placemat is a circle whose radius is equal to the diameter of the coaster. About how many square inches of the placemat are NOT covered by the coaster? Show your work.
area of large circle = π(49)
area of coaster = π(3.5^2) = 12.25π
The area not covered is the difference between the two areas.
To find the area of the placemat that is not covered by the coaster, we first need to find the area of the placemat and then subtract the area of the coaster.
The placemat is a circle whose radius is equal to the diameter of the coaster, which is 7 inches. Therefore, the radius of the placemat is also 7 inches.
To find the area of the placemat, we use the formula for the area of a circle: A = πr^2. Plugging in the value for the radius (7 inches):
A_placemat = π(7 inches)^2
=π(49 square inches)
= 49π square inches
So, the area of the placemat is 49π square inches.
Now, let's find the area of the coaster. The coaster is also a circle, and we can use the same formula to find its area. The diameter of the coaster is given as 7 inches, so the radius of the coaster is half of that, which is 3.5 inches.
A_coaster = π(3.5 inches)^2
=π(12.25 square inches)
= 12.25π square inches
So, the area of the coaster is 12.25π square inches.
To find the area of the placemat not covered by the coaster, we subtract the area of the coaster from the area of the placemat:
Area_not_covered = Area_placemat - Area_coaster
= 49π square inches - 12.25π square inches
= 36.75π square inches
Now, to estimate the value, we can use an approximation for π like 3.14:
Area_not_covered ≈ 36.75π square inches
≈ 36.75 * 3.14 square inches
≈ 115.245 square inches
Therefore, about 115.245 square inches of the placemat are not covered by the coaster.