The volume of a cone is 25/3 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)
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, we have V = 25/3π cm^3.
Let's find the value for the radius.
Since the cone's volume is given as (1/3)πr^2h, we have (1/3)πr^2h = 25/3π.
Canceling out the π's and multiplying both sides by 3, we get r^2h = 25.
The height of the cone is equal to the sphere's diameter, so we'll call it 2r (where r is the radius).
Substituting 2r for h in the equation r^2h = 25, we get r^2(2r) = 25.
Simplifying, we have 2r^3 = 25.
To find the value of r, we'll solve the equation:
2r^3 = 25
Divide both sides by 2: r^3 = 25/2
Take the cube root of both sides: r = ∛(25/2) = (25/2)^(1/3)
Now that we have the value for r, the radius of the sphere will be the same. We can now calculate the volume of the sphere.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
Substituting the value of r we found earlier, we have V = (4/3)π((25/2)^(1/3))^3.
Using a calculator, we can find the value of this expression:
V ≈ 65.449 cm^3.
Therefore, the volume of the sphere is approximately 65.449 cm^3.
in a fraction
To express the volume of the sphere as a fraction, we need to simplify the expression further.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
Substituting the value of r we found earlier, we have V = (4/3)π((25/2)^(1/3))^3.
Simplifying the expression inside the parentheses:
(25/2)^(1/3) = ∛(25/2) = (∛25)/(∛2) = (5)/(∛2).
Substituting this value back into the volume formula:
V = (4/3)π[(5)/(∛2)]^3
Raising a fraction to a power means raising both the numerator and denominator to that power:
V = (4/3)π(125/(∛2)^3)
V = (4/3)π(125/2^(1/3))^3
Simplifying further:
V = (4/3)π * 125^3/(2^(1/3))^3
V = (4/3)π * 125^3/(2^(1))
V = (4/3)π * 125^3/2
Raising 2 to the power of 1 gives 2:
V = (4/3)π * 125^3/2
V = (4/3)π * 125^3/2
Let's calculate this value:
V = (4/3) * π * (125^3/2)
V = (4/3) * π * (125^3) / (2)
V = (4/3) * π * 1953125 / 2
V = 812500π / 3
Therefore, the exact volume of the sphere is (812500π / 3) cm^3.
To find the volume of a sphere with the given conditions, we can follow these steps:
1. Calculate the radius of the cone:
The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where V is the volume, π is pi, r is the radius, and h is the height.
Since we are given the volume of the cone as 25/3 * π cm^3, we can rearrange the formula to solve for r.
25/3 * π = (1/3) * π * r^2 * h
25/3 = (1/3) * r^2 * (2r)
Simplifying further:
25 = r^3
Taking the cube root of both sides:
r = ∛25
r ≈ 2.92 cm (rounded to two decimal places)
2. Calculate the diameter of the sphere:
We know that the height of the cone is equal to the diameter of the sphere. Since the height of the cone is 2r, the diameter of the sphere is 2 * 2.92 cm = 5.84 cm.
3. Calculate the volume of the sphere:
The formula for the volume of a sphere is V = (4/3) * π * r^3, where V is the volume and r is the radius.
Since the radius of the sphere is the same as the radius of the cone (2.92 cm), we can use this value to calculate the volume:
V = (4/3) * π * (2.92 cm)^3
Calculating the volume:
V ≈ 143.67 cm^3 (rounded to two decimal places)
Therefore, the volume of the sphere is approximately 143.67 cm^3.