A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in Fig. 12.8. The height of the entire rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 cm. 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 $\pi = 3.14$)

Step 1: Identify Dimensions of Conical Part

We are given the height of the conical part, $h_c = 6\text{ cm}$. The diameter of the base of the conical portion is $5\text{ cm}$. To find the radius, we divide the diameter by $2$, so $r_c = 2.5\text{ cm}$.

Step 2: Calculate Slant Height of Cone

To calculate the curved surface area of the cone, we need its slant height ($l$). We can find the slant height using the Pythagorean theorem, where $l = \sqrt{r_c^2 + h_c^2}$. Substituting the values, we find $l = 6.5\text{ cm}$.

Step 3: Calculate Area Painted Orange (Conical Part)

The orange painted area includes the curved surface area of the cone ($\pi r_c l$) and the portion of the conical base that is visible. Since the cone is mounted on the cylinder, the visible part of the cone's base is an annulus. It's the area of the cone's base minus the area of the cylinder's base. However, the problem specifies the base of the conical portion has a diameter of 5 cm and the base of the cylindrical portion is 3 cm. This implies the conical part sits on top of the cylinder, so the top circular area of the cylinder will be covered by the cone. The visible orange part will be the curved surface area of the cone plus the annulus created by the cone base and the cylinder top. The radius of the cylinder $r_{cyl} = \frac{3}{2} = 1.5\text{ cm}$. The visible part of the base of the cone is $\pi r_c^2 - \pi r_{cyl}^2$. Adding these areas gives the total orange area.

Step 4: Identify Dimensions of Cylindrical Part

The total height of the rocket is $26\text{ cm}$, and the height of the conical part is $6\text{ cm}$. Therefore, the height of the cylindrical part ($h_{cyl}$) is $26 - 6 = 20\text{ cm}$. The base diameter of the cylindrical portion is $3\text{ cm}$, so its radius ($r_{cyl}$) is $1.5\text{ cm}$.

Step 5: Calculate Area Painted Yellow (Cylindrical Part)

The yellow painted area consists of the curved surface area of the cylinder and the area of its base. The curved surface area of a cylinder is $2\pi r_{cyl} h_{cyl}$. The area of the base is $\pi r_{cyl}^2$. Summing these two gives the total yellow area.

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