Question
Question 2: Use the relationship among formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cubic centimeters. What is the volume of a sphere if its radius is the same as the cylinder's and the height is equal to the sphere's diameter?
a. 18 cubic centimeters
b. 24 cubic centimeters
c. 54 cubic centimeters
d. 12 cubic centimeters
a. 18 cubic centimeters
b. 24 cubic centimeters
c. 54 cubic centimeters
d. 12 cubic centimeters
Answers
Answer
One of the answers is correct
Answered by
GPT 3.5
The volume of a cylinder is given by the formula Vcylinder = πr^2h, where r is the radius and h is the height.
The volume of a sphere is given by the formula Vsphere = (4/3)πr^3, where r is the radius.
Given that the volume of the cylinder is 36 cubic centimeters, we can set up the equation:
36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can rewrite the equation as:
36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides of the equation by 2π:
18 = r^3
Taking the cube root of both sides:
r = 3
Now we can substitute the value of r in the formula for the volume of the sphere:
Vsphere = (4/3)π(3)^3
Vsphere = (4/3)π(27)
Vsphere = 36π
Therefore, the volume of the sphere is 36π, which is approximately equal to 113.097 cubic centimeters.
None of the provided answer choices match this value, so the correct answer is not given.
The volume of a sphere is given by the formula Vsphere = (4/3)πr^3, where r is the radius.
Given that the volume of the cylinder is 36 cubic centimeters, we can set up the equation:
36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can rewrite the equation as:
36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides of the equation by 2π:
18 = r^3
Taking the cube root of both sides:
r = 3
Now we can substitute the value of r in the formula for the volume of the sphere:
Vsphere = (4/3)π(3)^3
Vsphere = (4/3)π(27)
Vsphere = 36π
Therefore, the volume of the sphere is 36π, which is approximately equal to 113.097 cubic centimeters.
None of the provided answer choices match this value, so the correct answer is not given.
Answered by
GPT 3.5
Apologies for the confusion. Let's re-evaluate the problem.
Given that the volume of the cylinder is 36 cubic centimeters, we can set up the equation:
Vcylinder = 36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can express h as 2r:
Vcylinder = 36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides of the equation by 2π:
18 = r^3
Taking the cube root of both sides:
r = 3
Now we can substitute the value of r in the formula for the volume of the sphere:
Vsphere = (4/3)π(3)^3
Vsphere = (4/3)π(27)
Vsphere = 36π
Therefore, the volume of the sphere is 36π, which is approximately equal to 113.097 cubic centimeters.
Among the given answer choices:
a. 18 cubic centimeters
b. 24 cubic centimeters
c. 54 cubic centimeters
d. 12 cubic centimeters
None of the choices match the calculated volume of 36π. Hence, there seems to be an error in the provided answer choices.
Given that the volume of the cylinder is 36 cubic centimeters, we can set up the equation:
Vcylinder = 36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can express h as 2r:
Vcylinder = 36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides of the equation by 2π:
18 = r^3
Taking the cube root of both sides:
r = 3
Now we can substitute the value of r in the formula for the volume of the sphere:
Vsphere = (4/3)π(3)^3
Vsphere = (4/3)π(27)
Vsphere = 36π
Therefore, the volume of the sphere is 36π, which is approximately equal to 113.097 cubic centimeters.
Among the given answer choices:
a. 18 cubic centimeters
b. 24 cubic centimeters
c. 54 cubic centimeters
d. 12 cubic centimeters
None of the choices match the calculated volume of 36π. Hence, there seems to be an error in the provided answer choices.
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