Question

The regular octagon in the finding values of sum and difference functions h(x) = x² - 1, k(x) = x – 2. (h k)(2) = (h – k)(3) = evaluate 3h(2) - 2k(3). The ceiling of this cathedral has a radius of 10.5 feet and a perimeter of 64 feet. What is the length of the apothem of the octagon? Round your answer to the nearest tenth of a foot.

Answer 1

The apothem of the regular octagon is approximately 9.7 feet, calculated using the Pythagorean theorem on the derived right triangle formed by the radius, a side of the octagon, and the apothem.

Step-by-step explanation:

The task is to calculate the length of the apothem of a regular octagon with a given radius and perimeter. We will use the fact that in a regular octagon, the radius forms a right triangle with the apothem and half of one side of the octagon. The perimeter of a regular octagon is the total length of all its sides, so if the perimeter is 64 feet, then one side (let's call it s) is 64 feet divided by 8, which equals 8 feet. The apothem (let's call it a) can be found using Pythagorean theorem where the hypotenuse is the radius (r = 10.5 feet), the one leg is half of the side (s/2 = 4 feet), so the apothem a is calculated as √(r^2 - (s/2)^2). After substituting the values, √(10.5^2 - 4^2) = √(110.25 - 16) = √(94.25) ≈ 9.7 feet. Hence, the apothem of the octagon is approximately 9.7 feet.