If you are calculating surface area of a right circular cone topped with a hemisphere. the radius is 4 cm and the rt angle is 8 cm...what is the lateral surface area. Is it 8.94 or is it 9? Explain answer please. Thanks
" the rt angle is 8 cm" makes no sense to me.
To calculate the lateral surface area of a right circular cone topped with a hemisphere, we need to find the curved surface area of the cone and add it to the curved surface area of the hemisphere.
First, let's find the curved surface area of the cone. The formula for the curved surface area of a cone is given as πrℓ, where r is the radius and ℓ is the slant height.
In this case, the radius of the cone is given as 4 cm. We need to find the slant height (ℓ). Since we know the right angle is 8 cm, we can use the Pythagorean theorem to find ℓ. The Pythagorean theorem states that the square of the hypotenuse (ℓ) is equal to the sum of the squares of the other two sides (radius and height).
Using the Pythagorean theorem:
ℓ^2 = r^2 + h^2
(8 cm)^2 = (4 cm)^2 + h^2
64 cm^2 = 16 cm^2 + h^2
48 cm^2 = h^2
h ≈ 6.93 cm
Now that we know the slant height, we can calculate the curved surface area of the cone:
Curved surface area of the cone = πrℓ
Curved surface area of the cone = π(4 cm)(6.93 cm)
Curved surface area of the cone ≈ 87.92 cm^2
Next, let's calculate the curved surface area of the hemisphere. The formula for the curved surface area of a hemisphere is given as 2πr^2.
In this case, the radius of the cone is also the radius of the hemisphere, which is given as 4 cm.
Curved surface area of the hemisphere = 2πr^2
Curved surface area of the hemisphere = 2π(4 cm)^2
Curved surface area of the hemisphere ≈ 100.53 cm^2
Finally, we can calculate the lateral surface area by adding the curved surface area of the cone and the curved surface area of the hemisphere:
Lateral surface area = Curved surface area of the cone + Curved surface area of the hemisphere
Lateral surface area ≈ 87.92 cm^2 + 100.53 cm^2
Lateral surface area ≈ 188.45 cm^2
Therefore, the lateral surface area is approximately 188.45 cm^2, not 8.94 or 9.