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 π=722)
Get the complete, step-by-step math solution for: "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...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 cm by 2 to find the radius of the hemispherical and conical parts, which is 1.75 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 cm and the radius r=1.75 cm into the equation for the height of the cone, h=H−r. This gives us h=3.25 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 cm and cone height h=3.25 cm into the formula for slant height. Square both values, add them, and then take the square root to get l≈3.691 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 2πr2.
Step 8: Calculate hemispherical CSA value
Substitute π=722 and r=1.75 cm into the formula for the CSA of the hemisphere. Perform the multiplication to get 19.25 cm2.
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 πrl, where l is the slant height.
Step 10: Calculate conical CSA value
Substitute π=722, r=1.75 cm, and l≈3.691 cm into the formula for the CSA of the cone. This calculation yields approximately 19.3327 cm2.
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 cm2) and the curved surface area of the cone (19.3327 cm2) to get the total area to be coloured. The total area is approximately 38.58 cm2.