40. Find the surface area of a conical grain storage tank that has a height of 42 meters and a diameter of 20 meters. Round the answer to the nearest square meter. (1 point)
3,028 m2
3,971 m2
1,357 m2
1,671 m2
To find the surface area of a conical grain storage tank, you need to calculate the curved surface area of the cone and the base area.
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.
To find the surface area of a conical grain storage tank, you need to calculate the lateral surface area and the base area separately, and then sum them up.
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.
post it.
As = pi*r^2 + pi*D*h
As = 3.14*100 + 3.14*20*42 = 2953 m^2.