A toy is in the shape of a right circular cylinder with hemisphere on one end and a cone on the other . The height and radius of the cylindrical part are 13 cm and 5 cm respectively. The radius of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if the height of the conical point is 12 cm
To calculate the surface area of the toy, we need to calculate the surface areas of the cylindrical, hemispherical, and conical parts separately, and then add them together.
1. Surface area of the cylindrical part:
The surface area of a right circular cylinder can be calculated using the formula: 2πrh, where r is the radius and h is the height.
In this case, the radius (r) of the cylindrical part is given as 5 cm and the height (h) is given as 13 cm. So, the surface area of the cylindrical part can be calculated as follows:
Surface area of cylindrical part = 2π(5)(13) = 130π cm²
2. Surface area of the hemispherical part:
The surface area of a hemisphere can be calculated using the formula: 2πr², where r is the radius.
In this case, the radius (r) of the hemispherical part is also given as 5 cm. So, the surface area of the hemispherical part can be calculated as follows:
Surface area of hemispherical part = 2π(5)² = 50π cm²
3. Surface area of the conical part:
The surface area of a cone can be calculated using the formula: πr(r + √(r² + h²)), where r is the radius and h is the height.
In this case, the radius (r) of the conical part is given as 5 cm and the height (h) is given as 12 cm. So, the surface area of the conical part can be calculated as follows:
Surface area of conical part = π(5)(5 + √(5² + 12²)) = 65π cm²
Now, we can add the surface areas of all three parts to get the total surface area of the toy:
Total surface area = Surface area of cylindrical part + Surface area of hemispherical part + Surface area of conical part
Total surface area = 130π + 50π + 65π
Total surface area = 245π cm²
To calculate the surface area of the toy, we need to find the surface area of each individual component and then add them together.
1. Surface area of the cylindrical part:
The formula for the surface area of a cylinder is given by:
Surface area = 2πr(r + h)
where r is the radius and h is the height of the cylinder.
In this case, r = 5 cm and h = 13 cm.
Using the formula, the surface area of the cylindrical part is:
Surface area of cylindrical part = 2π(5)(5 + 13) cm²
2. Surface area of the hemispherical part:
The formula for the surface area of a hemisphere is given by:
Surface area = 2πr²
where r is the radius of the hemisphere.
In this case, the radius of the hemisphere is also 5 cm.
Using the formula, the surface area of the hemispherical part is:
Surface area of hemispherical part = 2π(5)(5) cm²
3. Surface area of the conical part:
The formula for the surface area of a cone is given by:
Surface area = πr(r + √(r² + h²))
where r is the radius and h is the slant height of the cone.
In this case, r = 5 cm and h = 12 cm.
Using the formula, the surface area of the conical part is:
Surface area of conical part = π(5)(5 + √(5² + 12²)) cm²
4. Total surface area of the toy:
To find the total surface area of the toy, we add the surface areas of the three components together:
Total surface area = Surface area of cylindrical part + Surface area of hemispherical part + Surface area of conical part
Substituting the values calculated in previous steps into this equation will give us the final answer.