What is the surface area and volume of a composite figure made up of a hemisphere and a cone in terms of pi? The radius is 10, and the slant height is 15.
Ans: surface area: 350 pi units^2
volume: 1039.4 pi units^3
cone:
area = πrs
volume = 1/3 πr^2h = π/3 r^2√(s^2-r^2)
hemisphere:
area = 2πr^2
volume = 2/3 πr^3
So, plug in your numbers and you will get the desired answers. If not, come back with your work and we can see where you went wrong.
To find the surface area and volume of the composite figure made up of a hemisphere and a cone, we need to calculate separately for each shape and then add them together.
Let's start with the hemisphere:
The formula for the surface area of a hemisphere is:
Surface Area(H) = 2πr^2, where r is the radius.
Plugging in the given radius of 10, we get:
Surface Area(H) = 2π(10)^2 = 200π
The formula for the volume of a hemisphere is:
Volume(H) = (2/3)πr^3, where r is the radius.
Plugging in the given radius of 10, we get:
Volume(H) = (2/3)π(10)^3 ≈ 2093.3π
Now let's move on to the cone:
The formula for the slant height of a cone is:
slant height = √(r^2 + h^2), where r is the radius and h is the height.
We are given the slant height as 15 and the radius as 10. So we can calculate the height of the cone using the formula:
15 = √(10^2 + h^2)
Simplifying the equation, we get:
225 = 100 + h^2
h^2 = 225 - 100 = 125
h = √125
h ≈ 11.18
Now we can calculate the surface area and volume of the cone:
The formula for the surface area of a cone is:
Surface Area(C) = πr(r + √(r^2 + h^2)), where r is the radius and h is the height.
Plugging in the given radius of 10 and height of 11.18, we get:
Surface Area(C) = π(10)(10 + √(10^2 + 11.18^2)) ≈ 150π
The formula for the volume of a cone is:
Volume(C) = (1/3)πr^2h, where r is the radius and h is the height.
Plugging in the given radius of 10 and height of 11.18, we get:
Volume(C) = (1/3)π(10)^2(11.18) ≈ 372.1π
Finally, we can calculate the surface area and volume of the composite figure by adding the surface areas and volumes of the hemisphere and cone:
Surface Area(composite) = Surface Area(H) + Surface Area(C) = 200π + 150π = 350π
Volume(composite) = Volume(H) + Volume(C) = 2093.3π + 372.1π ≈ 1039.4π
Therefore, the surface area of the composite figure is 350π units^2, and the volume is approximately 1039.4π units^3.
To find the surface area and volume of a composite figure made up of a hemisphere and a cone, we first need to find the individual surface areas and volumes of each shape, and then add them together.
Let's start with the hemisphere. The surface area of a hemisphere is given by the formula:
SA_hemisphere = 2πr²
where r is the radius.
Given that the radius is 10, we can calculate the surface area of the hemisphere:
SA_hemisphere = 2π(10)² = 2π(100) = 200π square units.
Next, we move on to the cone. The surface area of a cone is given by the formula:
SA_cone = πr(r + l)
where r is the radius and l is the slant height.
Given that the radius is 10 and the slant height is 15, we can calculate the surface area of the cone:
SA_cone = π(10)(10 + 15) = π(10)(25) = 250π square units.
Now that we have the surface areas of the hemisphere and the cone, we can add them together to get the total surface area of the composite figure:
Total SA = SA_hemisphere + SA_cone = 200π + 250π = 450π square units.
Moving on to the volume, the volume of a hemisphere is given by the formula:
V_hemisphere = (2/3)πr³
Given that the radius is 10, we can calculate the volume of the hemisphere:
V_hemisphere = (2/3)π(10)³ = (2/3)π(1000) = (2000/3)π cubic units.
The volume of a cone is given by the formula:
V_cone = (1/3)πr²h
where r is the radius and h is the height.
Since the given figure is a cone with a slant height, we need to find the height of the cone using the Pythagorean theorem. The height is given by:
h = √(l² - r²)
where l is the slant height and r is the radius.
Given that the radius is 10 and the slant height is 15, we can calculate the height of the cone:
h = √(15² - 10²) = √(225 - 100) = √125 = 11.18 (approx.)
Now, we can calculate the volume of the cone:
V_cone = (1/3)π(10)²(11.18) = (1/3)π(100)(11.18) = (372.67)π cubic units.
Adding the volume of the hemisphere and the cone together gives us the total volume of the composite figure:
Total volume = V_hemisphere + V_cone = (2000/3)π + (372.67)π = (2572.67/3)π = 857.56π cubic units (approx.)
Therefore, the surface area of the composite figure is 450π square units, and the volume is 857.56π cubic units.
Cone surface area: π(10)(15)
= 150π
Hemisphere surface area: 2π(10)^2
= 200π
350π is the surface area
Cone volume: 1/3π(10)^2(11.2)
= 373.3π
I got the height by using the pythagorean theorem.
Hemisphere volume - 2/3π(10)^3
= 666.7π
= 1040π for the volume