Find the surface area of a right circular cone topped with a hemisphere. the height of the cone is 8cm, the radius is 4cm.

To find the surface area of a cone topped with a hemisphere, we first need to calculate the surface area of the cone and then add the surface area of the hemisphere.

1. Surface area of the cone:
The formula for the lateral surface area of a cone is given by A = πrl, where r is the radius of the base and l is the slant height of the cone.

Given:
Radius of the cone, r = 4 cm
Height of the cone, h = 8 cm

We can use the Pythagorean theorem to find the slant height, l:
l = √(r^2 + h^2)
l = √(4^2 + 8^2) = √(16 + 64) = √80 = 8√5 cm

The lateral surface area of the cone is:
A_cone = π * r * l = π * 4 * 8√5 = 32π√5 cm^2

2. Surface area of the hemisphere:
The formula for the surface area of a hemisphere is given by A = 2πr^2, where r is the radius of the hemisphere.

Given:
Radius of the hemisphere, r = 4 cm

The surface area of the hemisphere is:
A_hemisphere = 2π * r^2 = 2π * 4^2 = 32π cm^2

3. Total surface area:
To find the total surface area, we add the surface area of the cone and the surface area of the hemisphere:
Total surface area = A_cone + A_hemisphere
Total surface area = 32π√5 + 32π
Total surface area = 32π(√5 + 1) cm^2

Therefore, the surface area of the right circular cone topped with a hemisphere is 32π(√5 + 1) cm^2.

To find the surface area of a right circular cone topped with a hemisphere, we need to calculate the surface area of the cone and the hemisphere separately, and then add them together.

Let's start with the cone:

The surface area of a cone can be calculated using the formula:

SA_cone = π * r * ℓ + π * r^2

where r is the radius of the base of the cone and ℓ is the slant height.

Given that the radius of the cone is 4 cm, we need to find the slant height (ℓ). We can use the Pythagorean theorem to do that.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In our case, the slant height (ℓ) is the hypotenuse of a right triangle with one side being the height of the cone (8 cm) and the other side being the radius of the base (4 cm).

Using the Pythagorean theorem:

ℓ^2 = 8^2 + 4^2
ℓ^2 = 64 + 16
ℓ^2 = 80
ℓ = √80

Now we have the value of ℓ, we can substitute it into the formula for the surface area:

SA_cone = π * 4 * √80 + π * 4^2
SA_cone = 4π√80 + 16π

Moving on to the hemisphere:

The surface area of a hemisphere can be calculated using the formula:

SA_hemisphere = 2 * π * r^2

where r is the radius of the hemisphere.

Given that the radius of the cone is 4 cm, we can substitute it into the formula:

SA_hemisphere = 2 * π * 4^2
SA_hemisphere = 2 * π * 16
SA_hemisphere = 32π

Now, we can find the total surface area by adding the surface area of the cone and the hemisphere:

Total surface area = SA_cone + SA_hemisphere
Total surface area = 4π√80 + 16π + 32π

Simplifying the expression gives us the final answer for the surface area of the right circular cone topped with a hemisphere.