Two flat, circular plates are placed on a circular table top. The diameter of each plate is equal to the radius of the table top. Approximately how many square feet of the tabletop or not covered by the plates? Show or explain your work. Use 3.14 for pi
r of plate = r
2 r = R tabletop
area covered = 2 * pi r^2
total area of table = pi R^2 = pi (2r)^2 = 4 pi r^2
so area not covered = 2 pi r^2
r = Radius of the table.
r/2 = Radius of each plate.
At = pi * r^2. = Area of table top.
Ap = 2 * (pi * (r/2)^2) = 2 * (pi * r^2/4) = pi*r^2/2. = Area of 2 plates.
A = At - Ap = pi*r^2 - pi*r^2/2 = pi*r^2/2 = Area not covered.
So 1/2 of the area is not covered.
To find the area not covered by the plates on the tabletop, we first need to calculate the areas of the plates and the tabletop.
1. Area of each plate:
The diameter of each plate is equal to the radius of the tabletop, so the diameter of each plate is also d. The formula for the area of a circle is A = πr^2.
The radius of each plate = r (since the radius of the tabletop is equal to the diameter of the plate)
The area of each plate = πr^2
2. Area of the tabletop:
The tabletop is also a circle with a radius equal to r. So the formula for the area of the tabletop is the same as for the plates.
The radius of the tabletop = r
The area of the tabletop = πr^2
3. Calculating the area not covered by the plates:
We subtract the total area of the plates from the area of the tabletop to find the remaining uncovered area.
Uncovered area = Area of tabletop - (2 * Area of each plate)
Uncovered area = πr^2 - (2 * πr^2)
Uncovered area = πr^2 - 2πr^2
Uncovered area = πr^2(1 - 2)
Uncovered area = πr^2(-1)
Uncovered area = -πr^2
So, the area not covered by the plates on the tabletop is approximately -πr^2 square feet. However, it is not possible for the uncovered area to be negative, so the result is not valid. This indicates that there is either an error in the given information, or the uncovered area is equal to zero if the plates cover the entire tabletop.
To find out how many square feet of the tabletop are not covered by the plates, we need to compare the total area of the tabletop with the combined area of the plates.
Let's start by calculating the area of the tabletop. Since the diameter of each plate is equal to the radius of the tabletop, we can consider the diameter of the tabletop as twice the radius of the plate.
Let's represent the radius of the table as "r". Therefore, the diameter of the table is 2r.
The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.
So, the area of the tabletop is A_table = π(2r)^2 = 4πr^2.
Now, let's calculate the area of a single plate. Since the diameter of each plate is equal to the radius of the tabletop, the radius of each plate is r.
Therefore, the area of a single plate is A_plate = πr^2.
Since we have two plates, the combined area of the plates is 2A_plate = 2πr^2.
To find out how much of the tabletop is not covered by the plates, we subtract the combined area of the plates from the total area of the tabletop.
So, the area not covered by the plates is A_not_covered = A_table - 2A_plate.
Substituting the respective formulas, we get:
A_not_covered = 4πr^2 - 2πr^2.
Simplifying this expression, we get:
A_not_covered = 2πr^2.
Now, if we are given the value of the radius (r), we can calculate the area not covered by the plates. As an example, let's assume the radius is 5 feet.
A_not_covered = 2(3.14)(5^2).
A_not_covered = 2(3.14)(25).
A_not_covered = 157 square feet.
So, approximately 157 square feet of the tabletop are not covered by the plates.