Find the surface area of a conical grain storage tank that has a height of 30 meters and a of 16 meters. Round the answer to the nearest square meter.
Options:
2365 m squared
1762 m squared
981 m squared
780 m squared
r base = 8
slant height = sqrt (8^2 + 30^2) = sqrt (964) = 31.05
area (not including base) = pi r L = 3.14 * 8 * 31.05
= 780
if you include the base pi r^2 = 64 pi = 201
201 + 780 = 981
I do not know which you want. Probably just the sides because that is what you have to build out of metal.
16 is the diameter
To find the surface area of a conical grain storage tank, we need to calculate the lateral surface area and the base area separately and then add them together.
1. Lateral Surface Area:
The lateral surface area of a cone can be found using the formula: πrℓ, where ℓ represents the slant height of the cone.
Given that the height (h) of the cone is 30 meters, and the radius (r) is 16 meters, we can use the Pythagorean theorem to find the slant height (ℓ).
Using the Pythagorean theorem:
ℓ = √(r^2 + h^2)
ℓ = √(16^2 + 30^2)
ℓ = √(256 + 900)
ℓ = √1156
ℓ = 34 meters (approximately)
Now, let's calculate the lateral surface area:
Lateral surface area = πrℓ
Lateral surface area = π(16)(34)
Lateral surface area ≈ 1,701.94 square meters
2. Base Area:
The base area of a cone is given by the formula: πr^2, where r is the radius.
Base area = πr^2
Base area = π(16^2)
Base area = π(256)
Base area ≈ 804.25 square meters
Finally, we can find the total surface area by adding the lateral surface area and the base area:
Total surface area ≈ Lateral surface area + Base area
Total surface area ≈ 1,701.94 square meters + 804.25 square meters
Total surface area ≈ 2,506.19 square meters
Rounding the answer to the nearest square meter, the surface area of the conical grain storage tank is approximately 2,506 square meters.
Out of the given options, the closest answer is:
2365 m squared.
To find the surface area of a conical grain storage tank, we need to calculate the curved surface area and the base area, and then add them together.
First, let's calculate the curved surface area of the cone. The formula for the curved surface area of a cone is given by:
CSA = πrs
Where:
CSA = Curved Surface Area
r = radius of the cone's base
s = slant height of the cone
The radius (r) of the cone's base is half of the diameter. In this case, the diameter is 16 meters, so the radius is 8 meters.
The slant height (s) of the cone can be found using the Pythagorean theorem, where the height (h) is the vertical leg and the radius (r) is the horizontal leg.
Using the Pythagorean theorem:
s² = h² + r²
s² = 30² + 8²
s² = 900 + 64
s² = 964
s ≈ √964
s ≈ 31.048
Now that we have the radius (r = 8m) and the slant height (s ≈ 31.048m), we can substitute these values into the curved surface area formula:
CSA = π * 8 * 31.048
CSA ≈ 245.6 m²
Next, let's calculate the base area of the cone. The base area of a cone is given by:
Base Area = πr²
Substituting the value of the radius (r = 8m) into the formula:
Base Area = π * 8²
Base Area = π * 64
Base Area ≈ 201.06 m²
Finally, to find the surface area of the conical grain storage tank, we add the curved surface area and the base area:
Surface Area = CSA + Base Area
Surface Area ≈ 245.6 m² + 201.06 m²
Surface Area ≈ 446.66 m²
Rounding the answer to the nearest square meter, the surface area of the conical grain storage tank is approximately 447 m².
Therefore, none of the given options (2365 m², 1762 m², 981 m², 780 m²) match the correct answer.