Prove that for a cone made by rolling up a semicircle,the area of the curved surface is twice the base area

Question

A conical fire work is of base diameter 10 centimetres and height 12 centimetres. 10000 such fireworks are to be wrapped in colour paper. The price of the colour paper is 2 rupees per square metre. What is the total cost?

Answer 1

Find the total surface area

Base diameter is 10cm

Base radius r_b = 5cm

Hight h = 12cm.

By using Pythagoras theorem,

\(l=\sqrt{h^2+{r_b}^2}\)

\(=\sqrt{(12)^ 2+(5)^2}\)

\(=\sqrt {169}cm\)

= 13 cm

Total surface area of cone \(=\pi r_bl+\pi{r_b}^ 2\)

\(=\pi (5)(13)+\pi(5)^2\)

\(=65\pi +25 \pi\)

\(=282.6cm^ 2\)

Surface area of 10000 fire works is \(282.6(10000)=2826000cm^2.\)

\(28260000cm^2=\frac{2826000}{10000}\)

= 282.6cm²

Find the cost

Cost of the color paper is 282.6(2) = Rs. 565.20