Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses
cone = sphere – cylinder
cone = sphere – cylinder
cylinder = sphere – cone
cylinder = sphere – cone
cone = cylinder – sphere
cone = cylinder – sphere
sphere = cylinder + cone
sphere = cylinder + cone
Skip to navigation
The correct answer is: sphere = cylinder + cone.
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point)
Responses
54 cm3
54 cm cubed
18 cm3
18 cm cubed
12 cm3
12 cm cubed
24 cm3
To solve this problem, we need to find the volume of a sphere with the same radius as the cylinder.
The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. In this case, we are given that the volume of the cylinder is 36 cm³.
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius. We are trying to find the volume of the sphere.
Since the height of the cylinder is equal to the sphere's diameter, we can say that h = 2r.
Plugging in the given values, we have:
36 = πr²(2r)
36 = 2πr³
18 = πr³
Dividing both sides by π, we have:
r³ = 18/π
Now, we can find the volume of the sphere:
V = (4/3)πr³ = (4/3)π(18/π) = 24 cm³
Therefore, the volume of the sphere is 24 cm³.
The volume of a cone is 253 π cm3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
25 π cm3
To solve this problem, we need to find the volume of a sphere with the same radius as the cone.
The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height. In this case, we are given that the volume of the cone is 253π cm³.
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius. We are trying to find the volume of the sphere.
Since the height of the cone is equal to the sphere's diameter, we can say that h = 2r.
Plugging in the given values, we have:
253π = (1/3)πr²(2r)
253 = (2/3)πr³
759 = 2πr³
Dividing both sides by 2π, we have:
r³ = 759/(2π)
Now, we can find the volume of the sphere:
V = (4/3)πr³ = (4/3)π(759/(2π)) = (506/3)π
Therefore, the volume of the sphere is (506/3)π cm³, which is equivalent to 168.67π cm³.
Volume of Cones, Cylinders, and Spheres Quick Check
4 of 54 of 5 Items
Question
Use the image to answer the question.
A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.
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
36 in.3
36 in. cubed
27 in.3
27 in. cubed
18 in.3
18 in. cubed
40 in.3
To find the volume of the remaining portion after removing the cone from the cylinder, we need to subtract the volume of the cone from the volume of the cylinder.
Given that the volume of the cylinder is 54 in³, we can use the formula for the volume of a cylinder, V = πr²h, where r is the radius and h is the height.
Since the cone and cylinder share the same radius and height, the volume of the cone can be calculated using the formula for the volume of a cone, V = (1/3)πr²h.
Let's denote the radius of the cylinder and the cone as r and the height as h.
The volume of the cylinder is given as 54 in³, so we have:
54 = πr²h
We can also express the volume of the cone in terms of r and h:
V_cone = (1/3)πr²h
Substituting the value of h from the equation for the volume of the cylinder, we have:
V_cone = (1/3)πr²(54/πr²) = (54/3) in³ = 18 in³
Therefore, the volume of the remaining portion after removing the cone from the cylinder is 54 in³ - 18 in³ = 36 in³.
Hence, the correct answer is 36 in.³.
Use the image to answer the question.
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
18 m3
18 m cubed
24 m3
24 m cubed
36 m3
36 m cubed
There is not enough information to say.
There is not enough information to say.
Skip to navigation
To find the volume of the whole object, we need to calculate the sum of the volumes of the hemisphere and the cone.
Given that the volume of the cone is 12 m³, we can use the formula for the volume of a cone, V = (1/3)πr²h, where r is the radius and h is the height.
Given that the height of the cone is equal to the diameter of the hemisphere, we can say h = 2r, where r is the radius of the hemisphere.
Substituting this value into the volume formula for the cone, we have:
12 = (1/3)πr²(2r)
12 = (2/3)πr³
36 = 2πr³
Dividing both sides by 2π, we have:
r³ = 18/π
Now, let's calculate the volume of the hemisphere using the formula V = (2/3)πr³:
V_hemisphere = (2/3)π(18/π) = 12 m³
Finally, to find the volume of the whole object, we add the volume of the cone to the volume of the hemisphere:
V_object = V_cone + V_hemisphere = 12 m³ + 12 m³ = 24 m³
Therefore, the volume of the whole object is 24 m³.