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..
To find the surface area of the outside of the sculpture, we need to find the surface area of each cone separately and then add them together.
The formula for the surface area of a cone is given by A = πr(r + l), where r is the radius and l is the slant height.
For the first cone:
The radius is half of the diameter, so r = 8/2 = 4 feet.
The slant height is given as 5 feet.
Using the formula, the surface area of the first cone is A1 = 3.14 * 4(4 + 5) = 3.14 * 4(9) = 113.04 square feet.
For the second cone (which is the same as the first one):
The radius is the same as before, r = 4 feet.
The slant height is given as 5 feet.
Using the formula, the surface area of the second cone is A2 = 113.04 square feet.
Finally, we add the surface areas of the two cones together to get the total surface area of the outside of the sculpture:
Total surface area = A1 + A2 = 113.04 + 113.04 = 226.08 square feet.
Therefore, the designer needs approximately 226.08 square feet of steel for the outside of the sculpture.
A cone-shaped megaphone has a radius of 15 centimeters and a slant height of 20 centimeters. A megaphone has an open bottom. What is the lateral surface area of the megaphone, in square centimeters? Use 3.14 for pi.
The lateral surface area of a cone (including the slanted part but not including the base) is given by the formula A = π * r * l, where r is the radius and l is the slant height.
In this case, the radius is given as 15 centimeters and the slant height is given as 20 centimeters.
Using the formula, the lateral surface area of the megaphone is A = 3.14 * 15 * 20 = 942 square centimeters.
Therefore, the lateral surface area of the megaphone is 942 square centimeters.
To find out how many square feet of steel is needed for the outside of the sculpture, we first need to calculate the surface area of each cone. The formula to calculate the surface area of a cone is given by:
Surface Area = π * r * s
where:
- π is the mathematical constant approximately equal to 3.14,
- r is the radius of the base of the cone, and
- s is the slant height of the cone.
Now, let's calculate the surface area of each cone separately:
For the first cone:
Radius (r) = Diameter / 2 = 8 feet / 2 = 4 feet
Slant height (s) = 5 feet
Surface Area of Cone 1 = π * 4 feet * 5 feet
For the second cone:
Radius (r) = Diameter / 2 = 8 feet / 2 = 4 feet
Slant height (s) = 5 feet
Surface Area of Cone 2 = π * 4 feet * 5 feet
To find the total surface area of the hourglass sculpture, we add the surface areas of both cones:
Total Surface Area = Surface Area of Cone 1 + Surface Area of Cone 2
Total Surface Area = (π * 4 feet * 5 feet) + (π * 4 feet * 5 feet)
Now, we can calculate the total surface area:
Total Surface Area = 2 * π * 4 feet * 5 feet
Total Surface Area = 2 * 3.14 * 4 feet * 5 feet
Total Surface Area = 6.28 * 4 feet * 5 feet
Total Surface Area = 125.6 square feet
Therefore, the designer will need approximately 125.6 square feet of steel for the outside of the sculpture.