A designer is creating a modern art sculpture of an hourglass out of steel, to be on display at the town libraryThe hourglass is made by two cones meeting at the top pointThe designer wants to know how many square feet of steel 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
To calculate the surface area of the outside of the sculpture, we need to find the lateral area of each cone and add them together.
First, we need to find the slant height of each cone. Since the slant height is given as 5 feet, we can use the Pythagorean theorem to find the radius of each cone. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (slant height in this case) is equal to the sum of the squares of the other two sides.
Let's call the radius of the cone r and the height of the cone h.
The diameter is given as 8 feet, so the radius is 8/2 = 4 feet.
Using the Pythagorean theorem:
(5)^2 = r^2 + (4)^2
25 = r^2 + 16
r^2 = 25 - 16
r^2 = 9
r = √9
r = 3 feet
Now that we have the radius of the cone, we can find the lateral area of each cone. The lateral area of a cone is given by the formula: π * r * slant height.
For the first cone:
Lateral area = π * 3 * 5 = 15π square feet
For the second cone:
Lateral area = π * 3 * 5 = 15π square feet
Now, we can add the lateral areas of both cones together to find the total surface area of the outside of the sculpture.
Total surface area = 15π + 15π
= 30π square feet
Using the approximation π ≈ 3.14:
Total surface area ≈ 30 * 3.14
≈ 94.2 square feet
Therefore, approximately 94.2 square feet of steel is needed for the outside of the sculpture.