A designer is creating a modern art sculpture of an hourglass out of steel, to be on display at the town library. The hourglass is made by two cones meeting at the top point. The designer wants to know how many square feet of steel is needed for the outside of the sculpture, if the slant height of each cone is 5 feet and the diameter is 8 feet. Use 3.14 for pi (1 point)
326.56 square feet
653.12 square feet
226.08 square feet
113.04 square feet
To find the surface area of each cone, we first need to find the slant height of each cone. The slant height can be found using the Pythagorean theorem, which states that the square of the slant height is equal to the sum of the square of the radius and the square of the height.
Given that the diameter of each cone is 8 feet, the radius is half of that, which is 4 feet.
Using the Pythagorean theorem:
Slant height^2 = Radius^2 + Height^2
Slant height^2 = 4^2 + 5^2
Slant height^2 = 16 + 25
Slant height^2 = 41
Slant height = √41
The surface area of each cone can be found by adding the lateral surface area and the base area. The lateral surface area of a cone is given by the formula πrℓ, where r is the radius and ℓ is the slant height. The base area of a cone is given by the formula πr^2.
For each cone:
Lateral surface area = π * 4 * √41
Base area = π * 4^2
Adding these two areas together gives the total surface area for each cone:
Total surface area = (π * 4 * √41) + (π * 4^2)
To find the total surface area for both cones, we multiply this value by 2 (since there are two cones):
Total surface area = 2*((π * 4 * √41) + (π * 4^2))
Using 3.14 for π and simplifying the expression:
Total surface area = 2*(3.14 * 4 * √41 + 3.14 * 4^2)
Total surface area = 2*(3.14 * 4 * √41 + 3.14 * 16)
Total surface area = 2*(12.56 * √41 + 50.24)
Total surface area = 2*(12.56 * 6.40 + 50.24)
Total surface area = 2*(80.384 + 50.24)
Total surface area = 2*(130.624)
Total surface area = 261.248
Therefore, the number of square feet of steel needed for the outside of the sculpture is 261.248.
However, none of the given answer choices match this value, so none of the provided options are correct.