Find the surface area of a right circular cone topped with a hemisphere. the height of the cone is 8cm, the radius is 4cm.
I think we are looking at something like a single scoop icecream cone, right?
The surface area of a cone is pi(r)(s) where r is the radius and s is the slant height.
The surface area of a hemisphere
= (1/2)4pi(r^2) = 2pi(r^2)
we find s by using Pythagoras:
s^2 = 8^2 + 4^2 = 80
s = √80
So total surface area
= pi(4)√80 + 2pi(4^2)
= pi(16√5) + 32pi or appr. 212.93
check my arithmetic, its been horrible lately.
To find the surface area of a right circular cone topped with a hemisphere, you need to calculate the surface area of the cone and the surface area of the hemisphere separately and add them together.
Step 1: Calculate the surface area of the cone:
The surface area of a cone can be given by the formula:
A_cone = πr_1ℓ + πr_1^2
Where r_1 is the radius of the cone and ℓ is the slant height.
Given that the radius of the cone is 4cm, we need to find the slant height (ℓ).
Step 2: Calculate the slant height (ℓ):
The slant height can be found using the Pythagorean theorem:
ℓ = √(h^2 + r_1^2)
Where h is the height of the cone.
Given that the height of the cone is 8cm, we can calculate the slant height:
ℓ = √(8^2 + 4^2) = √(64 + 16) = √80 = 4√5 cm
Step 3: Calculate the surface area of the cone:
A_cone = π(4 cm)(4√5 cm) + π(4 cm)^2
A_cone = 16π√5 + 16π
A_cone = 16π(√5 + 1) cm^2
Step 4: Calculate the surface area of the hemisphere:
The surface area of a hemisphere can be given by the formula:
A_hemisphere = 2πr_2^2
Where r_2 is the radius of the hemisphere.
Given that the radius of the cone is 4cm, the radius of the hemisphere will also be 4cm.
Step 5: Calculate the surface area of the hemisphere:
A_hemisphere = 2π(4 cm)^2
A_hemisphere = 32π cm^2
Step 6: Calculate the total surface area:
The total surface area is the sum of the surface area of the cone and the surface area of the hemisphere.
A_total = A_cone + A_hemisphere
A_total = 16π(√5 + 1) cm^2 + 32π cm^2
A_total = 48π(√5 + 1) cm^2
Therefore, the surface area of the right circular cone topped with a hemisphere is 48π(√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 by calculating the surface area of the cone.
The formula to find the surface area of a cone is given by:
Surface Area of Cone = πrℓ + πr^2
Where:
- π is a mathematical constant (approximately 3.14159)
- r is the radius of the base of the cone
- ℓ is the slant height of the cone
In this case, you have given the height of the cone, which is 8cm, and the radius of the base, which is 4cm.
To find the slant height (ℓ), we can use the Pythagorean theorem:
ℓ = √(r^2 + h^2)
Substituting the values we have:
ℓ = √(4^2 + 8^2)
ℓ = √(16 + 64)
ℓ = √80
ℓ ≈ 8.94 cm (rounded to two decimal places)
Now we can calculate the surface area of the cone using the formula:
Surface Area of Cone = πrℓ + πr^2
Surface Area of Cone = π(4)(8.94) + π(4^2)
Surface Area of Cone = 35.57π + 16π
Surface Area of Cone ≈ 146.13 cm² (rounded to two decimal places)
Next, let's calculate the surface area of the hemisphere.
The formula to find the surface area of a hemisphere is given by:
Surface Area of Hemisphere = 2πr^2
Where:
- π is a mathematical constant (approximately 3.14159)
- r is the radius of the hemisphere
In this case, the radius of the hemisphere is also 4cm.
Surface Area of Hemisphere = 2π(4^2)
Surface Area of Hemisphere = 2π(16)
Surface Area of Hemisphere = 32π cm²
Now, we can add the surface area of the cone and the hemisphere to find the total surface area:
Total Surface Area = Surface Area of Cone + Surface Area of Hemisphere
Total Surface Area = 146.13 cm² + 32π cm²
Total Surface Area ≈ 146.13 cm² + 100.53 cm²
Total Surface Area ≈ 246.66 cm²
Therefore, the surface area of the right circular cone topped with a hemisphere is approximately 246.66 cm².