CONTEXT FOR PHOTO:A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3. What is the volume of the whole object?(1 point)
Responses
36 m^3
36 m cubed
24 m^3
24 m cubed
There is not enough information to say.
18 m^3
Please only use the responses as answer
24 m^3
The volume of the whole object can be calculated by adding the volume of the hemisphere and the volume of the cone together.
To find the volume of the cone, we can use the formula V = (1/3)πr^2h. Since the volume of the cone is given as 12 m^3 and the height is equal to the diameter of the hemisphere, we can substitute these values into the formula.
12 = (1/3)πr^2(d)
12 = (1/3)πr^2(2r)
12 = (2/3)πr^3
Simplifying further, we get:
36 = πr^3
r^3 = 36/π
r^3 ≈ 11.46
r ≈ 2.28
Now that we know the radius of the hemisphere, we can find its volume using the formula V = (2/3)πr^3.
V = (2/3)π(2.28)^3
V ≈ 31.01 m^3
Therefore, the volume of the whole object (hemisphere + cone) is approximately 31.01 m^3.
To find the volume of the whole object, which is composed of a hemisphere and a cone, we need to find the volume of each component separately and then add them together.
Let's start with the cone. We are given that the volume of the cone is 12 m^3. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the radius and h is the height of the cone.
In this case, the height is equal to the diameter of the hemisphere, which means that the radius of the cone is half the height. Let's represent the radius of the cone as r and the height as h. So, we have h = 2r.
Substituting this into the volume formula, we get 12 = (1/3) * π * r^2 * 2r.
Simplifying, we have 12 = (2/3) * π * r^3.
To find the value of r, we need the exact value of π. However, we can estimate π as 3.14 to get an approximate answer.
12 = (2/3) * 3.14 * r^3.
Rearranging the equation, we have r^3 = (12 * 3) / (2 * 3.14).
Solving for r, we find that r ≈ 1.91.
Now, we can find the height of the cone using h = 2r, which gives us h ≈ 3.82.
Now that we have the radius and height of the cone, we can calculate its volume. Plugging the values into the volume formula, we get V_cone = (1/3) * π * (1.91)^2 * 3.82 ≈ 14.62 m^3.
Next, we need to find the volume of the hemisphere. The volume of a hemisphere is given by V_hemisphere = (2/3) * π * r^3.
Since we already know the radius r, we can directly calculate the volume of the hemisphere using V_hemisphere = (2/3) * π * (1.91)^3 ≈ 10.43 m^3.
Finally, we can find the total volume of the object by adding the volume of the cone and the hemisphere:
V_total = V_cone + V_hemisphere ≈ 14.62 + 10.43 ≈ 25.05 m^3.
Therefore, the correct answer is 25.05 m^3.