5) A frustrum of a cone has the following dimensions: height = 6 in, lower radius = 8 in, upper radius = 4 inches.( Use pi= 3.14)
find each of the following:
5a) Lateral Surface area = _____ square inches
5b) Total Surface area= ______ square inches
5c) Volume = ______ cubic inches
pretend that you started with a large cone of height 12 in, and radius of 8 in.
To create the fulcrum, you cut off the top consisting of a cone of height 6 in and radius of 4 inches.
to find the lateral surface area of the fulcrum, find the lateral area of the large cone and subtract the lateral area of the imaginary cone that was cut off, leaving you with the lateral surface area of the fulcrum. Don't forget to add the area of the circle at the top of the fulcrum
for the total surface I would add the area of the circular base.
For the volume of the fulcrum, follow the same method
vol of fulcrum = vol of large cone - vol of cone cut off
To find the lateral surface area, total surface area, and volume of the frustrum cone, we can use the formulas associated with each.
5a) Lateral Surface area:
The lateral surface area of a frustrum cone can be found using the formula:
Lateral Surface Area = π × (lower radius + upper radius) × slant height
To find the slant height, we can use the Pythagorean theorem:
slant height = √((height^2) + ((lower radius - upper radius)^2))
Given: height = 6 inches, lower radius = 8 inches, and upper radius = 4 inches.
First, let's find the slant height:
slant height = √((6^2) + ((8 - 4)^2))
= √(36 + 16)
= √52
≈ 7.21 inches (rounded to two decimal places)
Now, we can calculate the lateral surface area:
Lateral Surface Area = 3.14 × (8 + 4) × 7.21
= 3.14 × 12 × 7.21
≈ 271.68 square inches (rounded to two decimal places)
Therefore, the lateral surface area of the frustrum cone is approximately 271.68 square inches.
5b) Total Surface area:
The total surface area of a frustrum cone can be found using the formula:
Total Surface Area = Lateral Surface Area + π × (lower radius^2) + π × (upper radius^2)
Given the values we already know:
Total Surface Area = 271.68 + (3.14 × (8^2)) + (3.14 × (4^2))
= 271.68 + (3.14 × 64) + (3.14 × 16)
= 271.68 + 200.96 + 50.24
≈ 523.88 square inches (rounded to two decimal places)
Therefore, the total surface area of the frustrum cone is approximately 523.88 square inches.
5c) Volume:
The volume of a frustrum cone can be found using the formula:
Volume = (1/3) × π × height × (lower radius^2 + upper radius^2 + (lower radius × upper radius))
Given the values we already know:
Volume = (1/3) × 3.14 × 6 × (8^2 + 4^2 + (8 × 4))
= (1/3) × 3.14 × 6 × (64 + 16 + 32)
= (1/3) × 3.14 × 6 × 112
≈ 706.08 cubic inches (rounded to two decimal places)
Therefore, the volume of the frustrum cone is approximately 706.08 cubic inches.