A cone is topped with a hemisphere.

Calculate the total surface area if the radius of the cone and hemisphere is
10 cm and the height of the cone is 24 cm.

hemisphere means "half a sphere"

Can you visualize the shape?

I see an "icecream cone with a single scoop on top"

in this case , use only the lateral surface of the cone + 1/2 the surface area of a sphere.

You have done these calculations in previous post.

To calculate the total surface area of the cone topped with a hemisphere, we need to find the areas of both the cone and the hemisphere and then add them together.

First, let's calculate the surface area of the cone. The formula for the surface area of a cone is given by:

Surface Area of Cone = π * r * (r + l)

where:
- π is a constant approximately equal to 3.14159
- r is the radius
- l is the slant height of the cone

We're given that the radius of the cone is 10 cm. To find the slant height (l), we can use the Pythagorean theorem:

l^2 = r^2 + h^2

where:
- h is the height of the cone

Substituting the values, we have:
l^2 = 10^2 + 24^2
l^2 = 100 + 576
l^2 = 676
l = √676
l = 26 cm (rounded to the nearest cm)

Now we can calculate the surface area of the cone:
Surface Area of Cone = π * r * (r + l)
Surface Area of Cone = 3.14159 * 10 * (10 + 26)
Surface Area of Cone = 3.14159 * 10 * 36
Surface Area of Cone = 1130.9734 cm^2 (rounded to 4 decimal places)

Next, let's calculate the surface area of the hemisphere. The formula for the surface area of a hemisphere is given by:

Surface Area of Hemisphere = 2 * π * r^2

We're given that the radius of the hemisphere is also 10 cm. Substituting the value, we have:
Surface Area of Hemisphere = 2 * 3.14159 * 10^2
Surface Area of Hemisphere = 2 * 3.14159 * 100
Surface Area of Hemisphere = 628.3185 cm^2 (rounded to 4 decimal places)

Finally, we can calculate the total surface area by adding the surface areas of the cone and the hemisphere together:
Total Surface Area = Surface Area of Cone + Surface Area of Hemisphere
Total Surface Area = 1130.9734 cm^2 + 628.3185 cm^2
Total Surface Area = 1759.2919 cm^2 (rounded to 4 decimal places)

Therefore, the total surface area of the cone topped with a hemisphere is approximately 1759.2919 cm^2.