The volume of a sphere depends on the sphere's radius:V=4/3πr^3.(A)If the radius of a sphere is 3 feet,what is the volume?Give exact and approximate answers and use correct units.(B)Solve for r in the volume formula.(C)If the volume of a sphere is 20 cubic inches,what is the radius?Give exact and approximate answers and use correct units.
(A) To find the volume of a sphere with a given radius, plug the value of the radius into the volume formula V = (4/3)πr^3.
Given that the radius is 3 feet, we can substitute r = 3 into the formula:
V = (4/3)π(3^3)
= (4/3)π(27)
= (4/3)(27π)
= 36π
So, the exact volume of this sphere is 36π cubic feet.
To get an approximate value, we can use the approximation π ≈ 3.14:
V ≈ 36 * 3.14
≈ 113.04
Therefore, the approximate volume is 113.04 cubic feet.
(B) To solve for r in the volume formula, we rearrange the formula by isolating r:
V = (4/3)πr^3
Multiply both sides of the equation by (3/4π) to cancel out the fraction:
(3/4π)V = r^3
Now, to isolate r, we take the cubic root of both sides:
r = (3/4π)V^(1/3)
So, the formula for r in terms of volume is r = (3/4π)V^(1/3).
(C) To find the radius of a sphere when given the volume, we can rearrange the volume formula V = (4/3)πr^3 and solve for r.
Given that the volume of the sphere is 20 cubic inches, we have:
20 = (4/3)πr^3
Divide both sides of the equation by (4/3)π:
(4/3)πr^3 = 20
Now, multiply both sides of the equation by 3/4π to isolate r^3:
r^3 = (20 * (3/4π))
r^3 = (15π)
To find the radius, take the cubic root of both sides:
r = (15π)^(1/3)
So, the exact radius of the sphere is (15π)^(1/3) inches.
To get an approximate value, we can use the approximation π ≈ 3.14:
r ≈ (15 * 3.14)^(1/3)
≈ 3.38
Therefore, the approximate radius is 3.38 inches.
actually
V = (4/3) pi r^3
if r = 3 ft
V = (4/3) pi (27) = 4 *9 * pi = 36 pi = 113.097 ft^3
(4/3) pi r^3 = 20 in^3
r^3 = (60/4 pi) = 15/ pi
r = (15/ pi)^1/3
= 1.684 inches