(a) A tent is in the shape of a right circular cylinder up to a height of 3 m and then a right circular cone, with a maximum height of 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of ₹ 2 per square metre, if the radius of the base is 14 m. OR (b) A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere of same radius. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy. Also, find the total surface area of the toy.

Step 1: Identify dimensions of the toy

The toy consists of a hemisphere and a cone. We are given the radius of the hemisphere, $r = 4.2\text{ cm}$, which is also the radius of the cone. The total height of the toy is $H = 10.2\text{ cm}$. The height of the conical part, $h$, can be found by subtracting the radius of the hemisphere from the total height.

Step 2: Calculate the slant height of the cone

To calculate the curved surface area of the cone, we need its slant height, $l$. We can find this using the Pythagorean theorem, as the radius, height, and slant height form a right-angled triangle. Substitute the values of $r$ and $h$ into the formula $l = \sqrt{r^2 + h^2}$.

Step 3: Calculate the volume of the toy

The total volume of the toy is the sum of the volume of the hemisphere and the volume of the cone. The formula for the volume of a hemisphere is $\frac{2}{3}\pi r^3$ and for a cone is $\frac{1}{3}\pi r^2 h$. We can factor out $\frac{1}{3}\pi r^2$ to simplify the calculation. Substitute the known values of $r$ and $h$ to find the total volume.

Step 4: Calculate the total surface area of the toy

The total surface area of the toy is the sum of the curved surface area of the hemisphere and the curved surface area of the cone. The formula for the curved surface area of a hemisphere is $2\pi r^2$ and for a cone is $\pi r l$. We can factor out $\pi r$ to simplify the calculation. Substitute the known values of $r$ and $l$ to find the total surface area.

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