A conical cavity is dug in a right circular cylinder of radius 2.1 CM and height 77 CM. If the height and base diameter of the cone are equal to that of the cylinder, find the cost... A conical cavity is dug in a right circular cylinder of radius 2.1 CM and height 77 CM. If the height and base diameter of the cone are equal to that of the cylinder, find the cost of tin plating it inside and outside at a rate of ₹1.40 per sq m. Read more

Steps to Solve

1. Identify the formulas for surface area For a cylinder, the total surface area $A_c$ is given by: $$ A_c = 2\pi r_h + 2\pi r^2 $$ where $r$ is the radius and $h$ is the height.

For a cone, the total surface area $A_{cone}$ is given by: $$ A_{cone} = \pi r (r + l) $$ where $r$ is the radius and $l$ is the slant height.

2. Substitute the dimensions into the formulas Let's say the cylinder has a radius of $r_c$ and a height of $h_c$, and the cone has a radius of $r_{cone}$ and a slant height of $l_{cone}$.

Calculate the total surface area of the cylinder: $$ A_c = 2\pi r_c h_c + 2\pi r_c^2 $$

Calculate the total surface area of the cone: First, calculate the slant height $l_{cone}$ using the Pythagorean theorem if necessary: $$ l_{cone} = \sqrt{h_{cone}^2 + r_{cone}^2} $$ Then substitute into: $$ A_{cone} = \pi r_{cone} (r_{cone} + l_{cone}) $$

3. Add the surface areas together Total surface area $A_{total}$ needed for plating is: $$ A_{total} = A_c + A_{cone} $$

4. Calculate the cost of tin plating If the cost per square unit of surface area is denoted as $C$, the total cost $Cost$ is calculated by: $$ Cost = A_{total} \times C $$

5. Final Calculations Perform all the calculations using the provided dimensions and cost to find the total cost.