To calculate the curved surface area and volume of the frustum, we need to understand some properties of a frustum of a cone.
(i) Curved Surface Area:
The curved surface area (CSA) of a frustum of a cone is calculated using the formula:
CSA = π(R + r) × l
Where:
- R and r are the radii of the top and bottom of the frustum, respectively,
- l is the slant height of the frustum.
To calculate the slant height (l), we can use the Pythagorean theorem:
l² = h² + (R - r)²
Given:
- R (radius of the top) = 3 cm
- r (radius of the bottom) = 7.5 cm
- h (height) = 30 cm
Let's substitute these values to calculate the slant height (l):
l² = 30² + (3 - 7.5)²
= 900 + (-4.5)²
= 900 + 20.25
= 920.25
Taking the square root of both sides, we get:
l = √920.25
≈ 30.34 cm (rounded to two decimal places)
Now we can calculate the curved surface area (CSA):
CSA = π(3 + 7.5) × 30.34
= 10.5π × 30.34
= 318.21 cm² (rounded to two decimal places)
Therefore, the curved surface area of the frustum is approximately 318.21 cm².
(ii) Volume:
The volume (V) of a frustum of a cone can be calculated using the formula:
V = (1/3)πh(R² + r² + Rr)
Given the same values as before, let's substitute them into the formula:
V = (1/3)π × 30 × (3² + 7.5² + 3 × 7.5)
= (1/3)π × 30 × (9 + 56.25 + 22.5)
= (1/3)π × 30 × 87.75
≈ 2757.75 cm³ (rounded to two decimal places)
Therefore, the volume of the frustum is approximately 2757.75 cm³.