Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere (see Fig 12.6). The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour. (Take $\pi = \frac{22}{7}$)

Step 1: Identify given dimensions and calculate radius

The problem provides the total height of the top and its diameter. We need to calculate the radius, which is half of the diameter. This radius is common to both the hemispherical and conical parts.

Step 2: Calculate radius value

We divide the given diameter of $3.5 \text{ cm}$ by 2 to find the radius of the hemispherical and conical parts, which is $1.75 \text{ cm}$.

Step 3: Calculate height of the conical part

The total height of the top ($H$) is the sum of the radius of the hemisphere and the height of the cone ($h$). Therefore, the height of the conical part can be found by subtracting the radius from the total height.

Step 4: Calculate cone height value

Substitute the total height $H = 5 \text{ cm}$ and the radius $r = 1.75 \text{ cm}$ into the equation for the height of the cone, $h = H - r$. This gives us $h = 3.25 \text{ cm}$.

Step 5: Calculate slant height of the cone

To find the curved surface area of the cone, we first need to determine its slant height ($l$). The slant height can be calculated using the Pythagorean theorem, as it forms the hypotenuse of a right-angled triangle with the cone's radius and height.

Step 6: Calculate slant height value

Substitute the calculated radius $r = 1.75 \text{ cm}$ and cone height $h = 3.25 \text{ cm}$ into the formula for slant height. Square both values, add them, and then take the square root to get $l \approx 3.691 \text{ cm}$.

Step 7: Calculate the curved surface area of the hemisphere

The area Rasheed has to colour includes the curved surface area of the hemispherical part. The formula for the curved surface area of a hemisphere is $2 \pi r^2$.

Step 8: Calculate hemispherical CSA value

Substitute $\pi = \frac{22}{7}$ and $r = 1.75 \text{ cm}$ into the formula for the CSA of the hemisphere. Perform the multiplication to get $19.25 \text{ cm}^2$.

Step 9: Calculate the curved surface area of the cone

The area to be coloured also includes the curved surface area of the conical part. The formula for the curved surface area of a cone is $\pi r l$, where $l$ is the slant height.

Step 10: Calculate conical CSA value

Substitute $\pi = \frac{22}{7}$, $r = 1.75 \text{ cm}$, and $l \approx 3.691 \text{ cm}$ into the formula for the CSA of the cone. This calculation yields approximately $19.3327 \text{ cm}^2$.

Step 11: Calculate total area to be coloured

The total area Rasheed has to colour is the sum of the curved surface area of the hemispherical part and the curved surface area of the conical part.

Step 12: Calculate total area value

Add the calculated curved surface area of the hemisphere ($19.25 \text{ cm}^2$) and the curved surface area of the cone ($19.3327 \text{ cm}^2$) to get the total area to be coloured. The total area is approximately $38.58 \text{ cm}^2$.

More from this chapter

• A playing top (lattu) is shaped like a cone surmounted by a hemisphere. The total height is 5 cm and the diameter is 3.5 cm. Find the surface area to be coloure
• Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed
• A toy is in the shape of a cone mounted on a hemisphere, both of radius 7 cm. The total height of the toy is 21 cm. Find the volume of the toy. (Use π = 22/7.)
• A decorative block is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a di
• Hello! I am your AI Math Tutor. Let's solve this problem step-by-step together. Let's start with Step 1! **Step 1: Visualize the cross-section** To solve this 3
• (a) The largest possible hemisphere is drilled out from a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the
• (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.
• A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the he

All Surface Areas and Volumes solutions →