Question

A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in figure. The height of the entire rocket is $26cm$, while the height of the conical part is $6cm$. The base of the conical portion has a diameter of $5cm$, while the base diameter of the cylindrical portion is $3cm$. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours (Take $\pi =3.14$)

Answer 1

Let the radius, slant height and height of the cone be $r,l,h$ and let the radius and height of the cylinder be $r^{\prime },h^{\prime }$ respectively.

Then $r=2.5cm,h=6cm,r^{\prime }=1.5cm,h^{\prime }=26-6=20cm$

$\Rightarrow l=\sqrt{r^2+h^2}=\sqrt{(2.5{)}^2+6^2}=6.5cm$

Here, the conical portion has its circular base resting on the base of the cylinder, but the base of the cone is larger than the base of the cylinder. So a part of the base the cone(a ring) is to be painted.

So the area to be painted orange $=$ Curved surface area of the cone $+$ base area of the cone $-$ base area of the cylinder.

$=\pi rl+\pi r^2-\pi (r^{\prime }{)}^2$

$=\pi \left[\right(2.5\times 6.5)+(2.5{)}^2-\left(1.5{)}^2\right]c{m}^2$

$=\pi \left[20.25\right]c{m}^2$

$=3.14\times 20.25c{m}^2$

$\therefore$ Area of the painted orange $=63.585c{m}^2$

Now, the area to be painted yellow $=$ Curved surface area of the cylinder $+$ area of one base of the cylinder

$=2\pi r^{\prime }{h}^{\prime }+\pi (r^{\prime }{)}^2$

$=\pi r^{\prime }[2h^{\prime }+r^{\prime }]$

$=(3.14\times 1.5)(2\times 20+1.5)c{m}^2$

$=4.71\times 41.5c{m}^2$

$\therefore$ Area to be painted yellow $=195.465c{m}^2$