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.
Are my answers right?
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
see the related questions below
To find the surface area of the composite figure made up of a hemisphere and a cone, you need to add the surface area of the cone and the surface area of the hemisphere.
To calculate the cone surface area, you correctly used the formula:
Cone surface area = πrs + πr^2
Where r is the radius of the base of the cone and s is the slant height.
So, the cone surface area would be:
Cone surface area = π(10)(15) + π(10)^2
= 150π + 100π
= 250π
For the hemisphere surface area, you correctly used the formula:
Hemisphere surface area = 2πr^2
So, the hemisphere surface area would be:
Hemisphere surface area = 2π(10)^2
= 200π
The total surface area of the composite figure would be the sum of the cone surface area and the hemisphere surface area:
Total surface area = Cone surface area + Hemisphere surface area
= 250π + 200π
= 450π
Therefore, the correct surface area of the composite figure is 450π, not 350π as you mentioned.
To find the volume of the composite figure, you need to calculate the volume of the cone and the volume of the hemisphere separately.
For the cone volume, you correctly used the formula:
Cone volume = (1/3)πr^2h
Where r is the radius of the base of the cone, and h is the height.
You mentioned that you used the Pythagorean theorem to find the height of the cone, which is correct. However, the slant height provided (15) is not sufficient to find the height accurately. To use the Pythagorean theorem, you need the height and the slant height of the cone.
For the hemisphere volume, you correctly used the formula:
Hemisphere volume = (2/3)πr^3
So, the hemisphere volume would be:
Hemisphere volume = (2/3)π(10)^3
= (2/3)π(1000/1)
= 666.67π
Therefore, the correct volume of the composite figure is 666.67π + Cone volume, not 1040π as you mentioned.
In conclusion, the correct surface area of the composite figure is 450π, and the correct volume is 666.67π + Cone volume. However, since the height of the cone was not provided, the exact volume cannot be calculated.
To calculate the surface area and volume of the composite figure made up of a hemisphere and a cone, you need to find the individual surface areas and volumes of each shape and then add them together.
Let's start with the cone:
Cone surface area = πr(rl)
= π(10)(15)
= 150π
Now let's move on to the hemisphere:
Hemisphere surface area = 2πr^2
= 2π(10)^2
= 200π
To find the total surface area of the composite figure, you need to add the surface areas of the cone and hemisphere:
Total surface area = Cone surface area + Hemisphere surface area
= 150π + 200π
= 350π
So, your answer of 350π for the surface area is correct.
Now let's calculate the volumes:
Cone volume = (1/3)πr^2h
= (1/3)π(10)^2(15)
= 150π
To find the height (h), you mentioned using the Pythagorean theorem. It would be helpful to provide the calculations you used.
Hemisphere volume = (2/3)πr^3
= (2/3)π(10)^3
= 666.7π
To find the total volume of the composite figure, you need to add the volumes of the cone and hemisphere:
Total volume = Cone volume + Hemisphere volume
= 150π + 666.7π
= 816.7π
Therefore, your answer of 816.7π for the volume is correct.
In summary, the surface area of the composite figure is 350π, and the volume is 816.7π.