A sculptor created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.3 , what is the volume of the amount remaining?(1 point) Responses 27 in.3 27 in. cubed 40 in.3 40 in. cubed 18 in.3 18 in. cubed 36 in.3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. Since the cylinder and cone share the same radius and height, the volume of the cylinder before removing the cone is 54 in.³.
We can represent the volume of the cone as Vcone = (1/3)πr^2h.
After removing the cone, the remaining volume will be the volume of the cylinder minus the volume of the cone: Vremaining = Vcylinder - Vcone.
Since the cylinder and cone have the same radius and height, we can substitute r and h into the equations:
Vcylinder = πr^2h = 54 in.³
Vcone = (1/3)πr^2h
Now we can substitute these values into the equation for Vremaining:
Vremaining = Vcylinder - Vcone
= 54 in.³ - (1/3)πr^2h
We notice that πr^2h is common in both terms, so we can factor it out:
Vremaining = πr^2h(1 - 1/3)
= πr^2h(2/3)
So, the volume of the remaining amount is given by 2/3 times the volume of the cylinder:
Vremaining = (2/3) * 54 in.³
= 36 in.³
Therefore, the volume of the remaining amount is 36 in.³.
To find the volume of the whole object, we need the dimensions of both the hemisphere and the cone. However, we are only given the volume of the cone, which is 12 m³.
Therefore, there is not enough information to determine the volume of the whole object.
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 There is not enough information to say. There is not enough information to say. 18 m3 18 m cubed 24 m3 24 m cubed 36 m3
To solve this problem, we need to calculate the volume of the removed cone and subtract it from the volume of the original cylinder.
The formula to calculate the volume of a cone is given by V = (1/3)πr^2h, where V represents the volume, π is a mathematical constant (approximately equal to 3.14159), r represents the radius of the base of the cone, and h represents the height of the cone.
In this case, the cone and cylinder share the same radius and height. Let's call the radius r and the height h.
To find the volume of the removed cone, we can use the formula V_cone = (1/3)πr^2h.
To find the volume of the original cylinder, we can use the formula V_cylinder = πr^2h.
The volume of the amount remaining is given by V_remaining = V_cylinder - V_cone.
Let's calculate the volume of the cone and the remaining volume step by step:
1. Calculate the volume of the cone:
V_cone = (1/3)πr^2h
2. Calculate the volume of the cylinder:
V_cylinder = πr^2h
3. Calculate the remaining volume:
V_remaining = V_cylinder - V_cone
Given that the volume of the original cylinder is 54 in.³, we need to solve for the unknowns.
Let's substitute the given information into the formulas and calculate the value:
V_cylinder = 54 in.³
πr^2h = 54 in.³
Since the radius and height are unknown, we cannot solve for them directly. Without additional information, it is not possible to determine the volume of the remaining amount.