As = pi*r^2 + pi*D*h
As = 3.14*100 + 3.14*20*42 = 2953 m^2.
3,028 m2
3,971 m2
1,357 m2
1,671 m2
As = 3.14*100 + 3.14*20*42 = 2953 m^2.
First, let's find the curved surface area of the cone. The formula for the curved surface area of a cone is given by:
CSA = π * r * l
where r is the radius of the base and l is the slant height of the cone.
Given that the diameter of the tank is 20 meters, the radius (r) is half of that, which is 20/2 = 10 meters.
To find the slant height (l) of the cone, we can use the Pythagorean theorem. The slant height, radius, and height of the cone form a right triangle. The height of the cone is given as 42 meters.
Using the Pythagorean theorem, we have:
l^2 = r^2 + h^2
l^2 = 10^2 + 42^2
l^2 = 100 + 1764
l^2 = 1864
l = √1864
l ≈ 43.17 meters
Now we can calculate the curved surface area (CSA):
CSA = π * 10 * 43.17
CSA ≈ 1362.89 m^2
Next, let's find the base area of the cone. The base of the cone is a circle, and the formula for the area of a circle is:
A = π * r^2
Given that the radius (r) is 10 meters, we can calculate the base area (A):
A = π * 10^2
A = π * 100
A ≈ 314.16 m^2
Finally, let's calculate the total surface area of the cone by adding the curved surface area and the base area:
Total surface area = CSA + A
Total surface area ≈ 1362.89 + 314.16
Total surface area ≈ 1677.05 m^2
Rounding to the nearest square meter, the surface area of the conical grain storage tank is approximately 1677 m^2.
Therefore, the answer is option D: 1,671 m^2.
First, let's find the lateral surface area. The formula for the lateral surface area of a cone is given by:
Lateral Surface Area = π * r * l
Where:
- π is a constant approximately equal to 3.14159
- r is the radius of the base of the cone (half the diameter)
- l is the slant height of the cone
To find the radius (r), divide the diameter by 2:
r = 20 m / 2 = 10 m
Next, we need to find the slant height (l). We can use the Pythagorean Theorem to accomplish this. The slant height (l) is the hypotenuse of a right triangle formed by the height (h) and the radius (r) of the cone. The formula is:
l = sqrt(h^2 + r^2)
Substituting the given values, we get:
l = sqrt(42^2 + 10^2) ≈ 42.64 m (rounded to two decimal places)
Now, we can calculate the lateral surface area:
Lateral Surface Area = π * r * l = 3.14159 * 10 m * 42.64 m ≈ 1339.54 m^2
Next, let's find the base area of the cone. The base area (A) of a cone is given by:
Base Area = π * r^2
So, the base area is:
Base Area = π * (10 m)^2 = 100π m^2 ≈ 314.159 m^2
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 = 1339.54 m^2 + 314.159 m^2 ≈ 1653.699 m^2
Rounding this answer to the nearest square meter, we get:
Surface Area ≈ 1654 m^2
Therefore, the correct answer is 1,654 m^2.