The volume of a hemisphere is 250
----- pi
3
Work out the total surface area of the solid hemisphere.
Give your answer as a multiple of pi.
Thank you.
V = 4/3 π r^3
A = 4π r^2 = 3V/r = ∛(36πV^2)
Now just plug in your numbers.
(4/3) π r^3 = 250π/3
4πr^3 = 250π
r^3 = 250/4 or 500/8
r = 500^(1/3) / 2
SA = 4π r^2
= 4π (500)^(2/3) / 8
= (1/2)(500)^(2/3) π
To find the total surface area of a solid hemisphere, we need to find the area of the curved surface and the area of the base.
The volume of a hemisphere is given by the formula:
V = (2/3) * pi * r^3
Given that the volume of the hemisphere is 250 * (pi/3), we can solve for the radius (r) by rearranging the formula:
250 * (pi/3) = (2/3) * pi * r^3
Canceling out the factors of (pi/3):
250 = 2 * r^3
Dividing both sides by 2:
125 = r^3
Taking the cube root of both sides:
r = 5
Now we have the radius (r = 5), and we can calculate the total surface area.
The curved surface area of a hemisphere is given by the formula:
A_c = 2 * pi * r^2
Substituting the value of the radius (r = 5):
A_c = 2 * pi * 5^2
= 2 * pi * 25
= 50 * pi
The area of the base of a hemisphere is given by the formula:
A_b = pi * r^2
Substituting the value of the radius (r = 5):
A_b = pi * 5^2
= pi * 25
= 25 * pi
To find the total surface area, we add the curved surface area and the area of the base:
Total surface area = A_c + A_b
= 50 * pi + 25 * pi
= 75 * pi
Therefore, the total surface area of the solid hemisphere is 75 * pi.
To find the total surface area of a solid hemisphere, we need to calculate the area of the curved surface (excluding the base) and the base.
1. The curved surface area of a hemisphere:
The curved surface area of a hemisphere is half the surface area of a sphere. Since we know the volume of the hemisphere, we can use the formula for the volume of a sphere to find its radius.
The volume of a sphere is given by:
V = (4/3) * π * r^3
Given that the volume of the hemisphere is 250π/3, we can set up the following equation:
250π/3 = (4/3) * π * r^3
Dividing both sides of the equation by (4/3)π, we get:
r^3 = (250π/3) / [(4/3)π] = 250 / 4 = 62.5
Taking the cube root of both sides, we find:
r = ∛(62.5) = 3.98 (approximately)
2. The curved surface area of a hemisphere:
The curved surface area of a hemisphere is given by:
CSA = 2 * π * r^2
Substituting the calculated value of r, we get:
CSA = 2 * π * (3.98)^2
Calculating this, we find:
CSA ≈ 99.81π (approximately)
3. The base area of a hemisphere:
The base area of a hemisphere is given by:
Base area = 2 * π * r^2
Substituting the calculated value of r, we get:
Base area = π * (3.98)^2
Calculating this, we find:
Base area ≈ 49.99π (approximately)
4. Total surface area of the hemisphere:
The total surface area is the sum of the curved surface area and the base area.
Total surface area = CSA + Base area
Total surface area ≈ 99.81π + 49.99π
Simplifying, we get:
Total surface area ≈ 149.8π
Therefore, the total surface area of the solid hemisphere is approximately 149.8π.