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 cubic meters. what is the volume of the whole object
To solve the problem, we can start by finding the volume of the cone using the given volume. The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
Since the height of the cone is equal to the diameter of the hemisphere, we can write the height of the cone as h = 2r. Substituting this into the formula for the volume of the cone, we have:
V_cone = (1/3)πr^2(2r) = (2/3)πr^3
Given that the volume of the cone is 12 cubic meters, we have the equation:
(2/3)πr^3 = 12
To find the radius "r", we can divide both sides of the equation by (2/3)π:
r^3 = 12 / ((2/3)π)
r^3 = 18/π
To solve for "r", we can take the cube root of both sides:
r = (18/π)^(1/3)
Now, we can find the volume of the hemisphere. The volume of a hemisphere is given by the formula V_hemisphere = (2/3)πr^3.
Substituting the value of "r" into the formula for the volume of the hemisphere, we have:
V_hemisphere = (2/3)π((18/π)^(1/3))^3
V_hemisphere = (2/3)π(18/π)
V_hemisphere = 12
Finally, we can find the total volume of the object by adding the volume of the hemisphere and the cone:
V_total = V_hemisphere + V_cone
V_total = 12 + 12
V_total = 24
Therefore, the volume of the whole object is 24 cubic meters. Hence, the correct answer is A. 24 cubic m.
To find the volume of the whole object, we need to find the volume of both the hemisphere and the cone and then add them together.
Let's assume that the radius of the hemisphere is "r" and the height of the cone is also "r" (as given).
The volume of the hemisphere can be calculated using the formula: V_hemisphere = (2/3)πr^3.
The volume of the cone can be calculated using the formula: V_cone = (1/3)πr^2h, where h is the height of the cone.
Since the height of the cone is equal to the diameter of the hemisphere, we can substitute "r" for "h" in the formula for the volume of the cone.
V_cone = (1/3)πr^2(r) = (1/3)πr^3
Given that the volume of the cone is 12 cubic meters, we have the equation:
(1/3)πr^3 = 12
To find the radius "r", we can multiply both sides of the equation by 3/(πr^2):
r^3 = (12 * (3/(πr^2)))
r^3 = 36/(πr^2)
r^5 = 36/π
To solve for "r", we can take the fifth root of both sides:
r = (36/π)^(1/5)
Now, we can substitute this value of "r" back into the formula for the volume of the hemisphere to find its volume:
V_hemisphere = (2/3)π(36/π)^(1/5)^3
Finally, we can add the volumes of the hemisphere and the cone to find the total volume of the object:
V_total = V_hemisphere + V_cone
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 cubic meters. what is the volume of the whole object
A. 24 cubic m
B. 36 cubic m
C. 18 cubic m
D. there is not enough information to say
To find the volume of the entire object, which is composed of a hemisphere and a cone, we need to determine the volumes of each component separately and then add them together.
First, let's find the volume of the cone. Given that the volume of the cone is 12 cubic meters, we have:
Volume of Cone = π * r^2 * h_cone / 3
Since the height (h_cone) of the cone is equal to the diameter of the hemisphere, we can write:
h_cone = 2 * r_hemisphere
Substituting this back into the volume formula, we get:
Volume of Cone = π * r^2 * (2 * r_hemisphere) / 3
Next, let's determine the volume of the hemisphere. The volume of a hemisphere is given by:
Volume of Hemisphere = 2 * π * r_hemisphere^3 / 3
Now, we can sum up the volume of the cone and the volume of the hemisphere to find the volume of the entire object:
Volume of Entire Object = Volume of Cone + Volume of Hemisphere
Substituting the respective formulas, we have:
Volume of Entire Object = (π * r^2 * (2 * r_hemisphere) / 3) + (2 * π * r_hemisphere^3 / 3)
By substituting the given value for the cone volume (12 cubic meters), we can solve for the remaining variables and compute the volume of the entire object.