The diameters of the top and bottom of a frustum of a cone are 6cm and
15cm respectively. If the height of the frustum is 30cm, calculate (i) the
curved surface area (ii) the volume of the frustum.
Note: solution needed
(i) Curved Surface Area = π × (R1 + R2) × √(R1 - R2)² + h²
= π × (6 + 15) × √(6 - 15)² + 30²
= π × 21 × √(-9) + 900
= π × 21 × 30
= 1890π cm²
(ii) Volume of the Frustum = (1/3) × π × h × (R1² + R1 × R2 + R2²)
= (1/3) × π × 30 × (6² + 6 × 15 + 15²)
= (1/3) × π × 30 × (36 + 90 + 225)
= (1/3) × π × 30 × 351
= 3510π cm³
To calculate the curved surface area and volume of a frustum of a cone, we can use the following formulas:
i) Curved Surface Area (CSA) of Frustum of Cone = π(R₁ + R₂) × l
ii) Volume of Frustum of Cone = (1/3)πh(R₁² + R₂² + R₁R₂)
where:
- R₁ and R₂ are the radii of the top and bottom circles of the frustum,
- h is the height of the frustum, and
- l is the slant height of the frustum.
Now, let's calculate the curved surface area and volume using the given dimensions.
i) Calculating the Curved Surface Area (CSA):
Given: R₁ = 6 cm, R₂ = 15 cm, h = 30 cm
First, we need to find the slant height (l) using the Pythagorean theorem:
l = √(h² + (R₂ - R₁)²)
Substituting the given values, we have:
l = √(30² + (15 - 6)²)
l = √(900 + 9²)
l = √(900 + 81)
l = √981
l ≈ 31.3 cm (rounded to one decimal place)
Now, substitute the values of R₁, R₂, and l into the CSA formula:
CSA = π(6 + 15) × 31.3
CSA = 21.2π cm² (rounded to one decimal place)
ii) Calculating the Volume:
Given: R₁ = 6 cm, R₂ = 15 cm, h = 30 cm
Substituting the given values into the volume formula:
Volume = (1/3)π × 30(6² + 15² + 6 × 15)
Volume = 1050π cm³ (rounded to one decimal place)
Therefore, the (i) curved surface area of the frustum is approximately 21.2π cm², and (ii) the volume of the frustum is approximately 1050π cm³.
To solve this problem, we can use the formulas for the curved surface area and volume of a frustum of a cone.
(i) Curved Surface Area:
The curved surface area of a frustum of a cone can be calculated using the formula:
CSA = π(R1 + R2) l
where R1 is the radius of the top (smaller) base, R2 is the radius of the bottom (larger) base, and l is the slant height.
Given that the diameters of the top and bottom of the frustum are 6cm and 15cm respectively, we can calculate the radii:
R1 = 6cm / 2 = 3cm
R2 = 15cm / 2 = 7.5cm
Since the height of the frustum is not given, we will need to calculate the slant height. Using the Pythagorean theorem, we have:
l^2 = h^2 + (R2 - R1)^2
Plugging in the given values:
l^2 = (30cm)^2 + (7.5cm - 3cm)^2
= 900cm^2 + 22.5cm^2
= 922.5cm^2
Taking the square root of both sides, we get:
l = √(922.5cm^2)
≈ 30.4cm
Now that we have the values of R1, R2, and l, we can substitute them into the formula for CSA:
CSA = π(R1 + R2) l
= π(3cm + 7.5cm) × 30.4cm
≈ 342.29cm^2
Therefore, the curved surface area of the frustum is approximately 342.29cm^2.
(ii) Volume:
The volume of a frustum of a cone can be calculated using the formula:
V = 1/3πh (R1^2 + R2^2 + R1R2)
In this case, the given height is 30cm and the radii are R1 = 3cm and R2 = 7.5cm. Substituting these values into the formula:
V = 1/3π × 30cm (3cm^2 + 7.5cm^2 + 3cm × 7.5cm)
≈ 2204.8cm^3
Therefore, the volume of the frustum is approximately 2204.8cm^3.