A circular tablecloth is draped over a rectangular table so that the center of the cloth is directly above the center of the table. The table is 6 X(times) 2sqrt(3)feet, and the cloth has a radius of 2 feet. What is the area of the portion of the tablecloth covered by the cloth?
Please proofread your questions before you post them.
6 * 2√3 = 6 * 2 * 1.7 = ?
A = πr^2 = 3.14 * 4 = ?
Solve and compare.
To find the area of the portion of the tablecloth covered by the cloth, we need to calculate the area of the circular portion that hangs down from the table.
First, let's calculate the diameter of the circular tablecloth. Since the table is 6 X 2sqrt(3) feet, the diagonal of the table can be found using the Pythagorean theorem:
diagonal = √(6^2 + (2sqrt(3))^2)
= √(36 + 12)
= √48
= 4√3 feet
The diameter of the tablecloth will be equal to the diagonal of the table, which is 4√3 feet. Therefore, the radius of the tablecloth is half of the diameter, which is 2√3 feet.
Now we know that the radius of the tablecloth is 2 feet. To find the area of the portion covered by the cloth, we need to calculate the area of the circular sector formed by the hanging portion of the tablecloth.
The formula to calculate the area of a circular sector is:
Area = (θ/360) * π * r^2
where θ is the central angle of the sector (in degrees) and r is the radius of the sector.
In this case, we need to find the central angle of the circular sector. The central angle can be calculated using the right-angled triangle formed by the center of the cloth, the radius of the cloth, and half the length of the table.
Using trigonometry:
cos(θ) = (half the length of the table) / (radius of the cloth)
cos(θ) = 3 / 2√3
θ = arccos(3 / (2√3))
θ ≈ 30.96 degrees
Now, we can substitute the values into the area formula:
Area = (θ/360) * π * r^2
= (30.96/360) * π * (2^2)
= (0.086 * 3.14 * 4)
≈ 1.08 square feet
Therefore, the area of the portion of the tablecloth covered by the cloth is approximately 1.08 square feet.