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 24x−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
16x−24cm3
16 x minus 24 cm cubed
8x−12 cm3
8 x minus 12 cm cubed
36x−54 cm3
36 x minus 54 cm cubed
12x−18 cm3
BOT, ITS WRONGG
To solve this problem, we need to use the relationships among the formulas for the volumes of cones, cylinders, and spheres.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height.
The formula for the volume of a sphere is V_sphere = (4/3)πr^3, where r is the radius.
Given that the volume of the cylinder is 24x - 36 cm^3 and the height of the cylinder is equal to the sphere's diameter, we can find the radius of the cylinder.
Since the height of the cylinder is equal to the diameter of the sphere, we can say that 2r = h.
Substituting this relationship into the volume formula for the cylinder, we get V_cylinder = πr^2(2r) = 2πr^3.
Now, we can set up an equation to find the radius of the cylinder:
2πr^3 = 24x - 36
Dividing both sides of the equation by 2π, we have:
r^3 = (24x - 36) / (2π)
r^3 = 12x - 18 / π
Taking the cube root of both sides, we get:
r = (12x - 18 / π)^(1/3)
Now that we have the radius of the sphere, we can substitute it into the formula for the volume of a sphere:
V_sphere = (4/3)πr^3
V_sphere = (4/3)π((12x - 18 / π)^(1/3))^3
V_sphere = (4/3)π(12x - 18 / π)
V_sphere = (4/3)(12x - 18)
V_sphere = 16x - 24 cm^3
Therefore, the volume of the sphere is 16x - 24 cm^3.
To solve this problem, we can use the formulas for the volume of a cylinder and the volume of a sphere:
Volume of a cylinder: Vcylinder = πr^2h
Volume of a sphere: Vsphere = (4/3)πr^3
The problem tells us that the volume of the cylinder is 24x-36 cm^3. This means that Vcylinder = 24x-36.
We also know that the height of the cylinder is equal to the sphere's diameter. Since the diameter is twice the radius, this means that the height of the cylinder is 2r.
To find the volume of the sphere, we need to find the radius. We can set the volume of the cylinder equal to Vcylinder = πr^2(2r), and solve for r:
24x-36 = πr^2(2r)
24x-36 = 2πr^3
Divide both sides by 2π:
12x-18 = r^3
Now, substitute this value of r into the formula for the volume of the sphere:
Vsphere = (4/3)πr^3
Vsphere = (4/3)π(12x-18)^3
Simplifying this expression, we get:
Vsphere = (4/3)π(1728x^3 - 972x^2 + 216x - 18)
Therefore, the volume of the sphere is 1728x^3 - 972x^2 + 216x - 18 cm^3.
Apologies for the incorrect response. Let me correct it:
The volume of a cylinder is given by the formula Vcylinder = πr^2h, where r is the radius and h is the height of the cylinder.
Given that the volume of the cylinder is 24x - 36 cm^3, we can equate this to the formula for the volume of a cylinder:
24x - 36 = πr^2h
We also know that the height of the cylinder is equal to the diameter of the sphere, which means h = 2r.
Substituting 2r for h in the equation:
24x - 36 = πr^2(2r)
Simplifying further:
24x - 36 = 2πr^3
Divide both sides by 2π:
12x - 18 = r^3
Now, let's find the volume of the sphere. The formula for the volume of a sphere is Vsphere = (4/3)πr^3.
Substituting r^3 = 12x - 18 into the formula:
Vsphere = (4/3)π(12x - 18)
Simplifying this expression gives us the volume of the sphere as:
Vsphere = (16/3)πx - 8π
Therefore, the correct answer is (16/3)πx - 8π cm^3.