Question

An oil funnel made of tin sheet consists of a cylindrical portion 10 cm long attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8 cm and the diameter of the top of funnel is 18 cm. Find the area of tin required to make the funnel.

Answer 1

1. Identify component dimensions

An oil funnel is a combination of two hollow shapes open at both ends:

Cylindrical portion length (h₁) = 10 cm.

Cylindrical portion radius (r₂) = `"Diameter"/2` = `8/2` = 4 cm.

Frustum of a cone height (h₂) = Total height – Cylinder height = 22 – 10 = 12 cm.

Frustum top radius (r₁) = `"Top diameter"/2` = `18/2` = 9 cm.

Frustum bottom radius (r₂) = Same as the cylinder radius = 4 cm.

2. Determine frustum slant height

To find the surface area of the cone frustum, calculate its slant height `(l)`:

`l = sqrt(h_2^2 + (r_1 - r_2)^2`

`l = sqrt(12^2 + (9 - 4)^2`

= `sqrt(144 + 5^2)` 

`l = sqrt(144 + 25)`

= `sqrt(169)`

= 13 cm

3. Calculate surface areas

The total area of tin required equals the sum of the Curved Surface Areas (CSA) of both parts:

1. CSA of the cylinder: `"CSA"_"cylinder"` = 2πr₂h₁

= 2 × π × 4 × 10

= 80π cm²

2. CSA of the frustum: `"CSA"_"frustum" = π(r_1 + r_2)l`

= π(9 + 4) × 13

= 13 × 13π

= 169π cm²

4. Sum the regions

Total area = 80π + 169π

= 249π cm²

Using `π ≈ 22/7`:

Total area = `249 xx 22/7`

= `5478/7 cm^2` ≈ 782.57 cm²