A village water tank is in the form of a frustrum of a cone of height 3.2 cm.The top and bottom radii are 18cm and 24cm respectively.Calculate:The surface area of the tank
A town water tank with dimensions in cm? I don't think so! But, using the numbers given,
since 18/24 = 3/4, the top 3/4 of the cone is missing.
So, the whole cone would have a height of 12.8, with a slant height of 27.2
The surface of the whole cone is πr(r+s) = 1228.8π
The area of the missing 3/4 is 9/16 the whole area, or 690.75π
Thus, the area of the frustrum is 538.05π + 324π = 862.05π
how did you get 12.8
To calculate the surface area of the tank, we need to consider the curved surface area of the frustum, as well as the top and bottom areas.
1. Curved Surface Area:
The curved surface area of a frustum of a cone can be calculated using the formula: A = π(r₁ + r₂)ℓ,
where:
- A is the curved surface area,
- r₁ is the radius of the top circle,
- r₂ is the radius of the bottom circle,
- ℓ is the slant height of the frustum.
To find the slant height (ℓ), we can use the Pythagorean theorem:
ℓ = √((h² + (r₂ - r₁)²),
where:
- h is the height of the frustum.
Given:
- r₁ = 18 cm (radius of the top circle),
- r₂ = 24 cm (radius of the bottom circle),
- h = 3.2 cm (height of the frustum).
Calculations:
ℓ = √((3.2²) + (24 - 18)²)
ℓ = √((10.24) + (6)²)
ℓ = √(10.24 + 36)
ℓ = √(46.24)
ℓ ≈ 6.80 cm (rounded to two decimal places)
Next, we can calculate the curved surface area:
A = π(18 + 24)(6.80)
A = π(42)(6.80)
A ≈ 897.16 cm² (rounded to two decimal places)
2. Top and Bottom Areas:
The top and bottom areas of the frustum are the areas of the circles formed by the top and bottom radii, respectively.
Area of the top circle = π(r₁)²
Area of the bottom circle = π(r₂)²
Calculations:
Area of the top circle = π(18)²
Area of the bottom circle = π(24)²
Area of the top circle ≈ 1017.88 cm² (rounded to two decimal places)
Area of the bottom circle ≈ 1809.56 cm² (rounded to two decimal places)
3. Total Surface Area:
To find the total surface area of the tank, we can add up the curved surface area, the top area, and the bottom area:
Total Surface Area = Curved Surface Area + Area of the Top Circle + Area of the Bottom Circle
Total Surface Area = 897.16 + 1017.88 + 1809.56
Total Surface Area ≈ 3724.60 cm² (rounded to two decimal places)
Therefore, the surface area of the tank is approximately 3724.60 cm².
To calculate the surface area of the tank, we need to find the area of the curved surface and the area of the top and bottom bases separately, and then add them together.
1. Area of the curved surface (lateral area) of the frustum of a cone:
The curved surface area of a frustum of a cone can be calculated using the formula:
A = π(R + r) × l
Where:
A = Curved surface area of the frustum
R = Radius of the larger base (24 cm)
r = Radius of the smaller base (18 cm)
l = Slant height of the frustum
To calculate the slant height (l), we can use the Pythagorean theorem:
l = √(h^2 + (R - r)^2)
Where:
h = Height of the frustum (3.2 cm)
R = Radius of the larger base (24 cm)
r = Radius of the smaller base (18 cm)
Substituting the values, we have:
l = √(3.2^2 + (24 - 18)^2)
l = √(10.24 + 36)
l = √46.24
l ≈ 6.8 cm
Now, substitute the values into the formula:
A = π(24 + 18) × 6.8
A = π(42) × 6.8
A ≈ 879.6 cm²
2. Area of the top and bottom bases:
The area of the circular base can be calculated using the formula:
A = πr²
Where:
A = Area of the base
r = Radius of the base
The top and bottom bases are circles with radii 18 cm and 24 cm, respectively. Therefore, their areas can be calculated as follows:
A_top = π(18)^2
A_top ≈ 1017.8 cm²
A_bottom = π(24)^2
A_bottom ≈ 1809.6 cm²
3. Total surface area of the tank:
Now, let's calculate the total surface area of the tank by adding the curved surface area and the area of the top and bottom bases:
Total surface area = Curved surface area + Area of the top base + Area of the bottom base
Total surface area ≈ 879.6 + 1017.8 + 1809.6
Total surface area ≈ 3707 cm²
Therefore, the surface area of the tank is approximately 3707 cm².