The volume of a sphere is given by V=4*r^3/3, where r is the radius,compute the radius of a sphere having a volume 40 percent greater than that of a sphere of radius 4 ft?
Volume at r = 4
= (4/3)π 4^3
= 256π/3 = appr 268.08...
that increased by 40% ----> 256π/3 * 1.4
= 256π/3(14/10)
= 3584π/30 = appr 375.32
then 3584π/30 = (4/3)π r^3
(3584/30)(3/4) = r^3
r^3 = 89.6
r = appr 4.47 ft
To find the radius of a sphere with a volume 40% greater than that of a sphere with radius 4 ft, we can follow these steps:
1. First, let's find the volume of the sphere with a radius of 4 ft.
- Given: r = 4 ft
- Using the formula V = (4/3) * π * r^3, where π is approximately 3.14159:
- V = (4/3) * 3.14159 * (4 ft)^3
- V = (4/3) * 3.14159 * 64 ft^3
- V = 268.082 ft^3
2. Next, let's find the desired volume, which is 40% greater than the volume of the sphere with a radius of 4 ft.
- Given: Desired volume = 40% greater than 268.082 ft^3
- Desired volume = 268.082 ft^3 + (0.4 * 268.082 ft^3)
- Desired volume = 268.082 ft^3 + 107.2328 ft^3
- Desired volume = 375.3148 ft^3
3. Now, we can plug the desired volume into the volume formula and solve for the radius of the sphere.
- Given: V = 375.3148 ft^3
- V = (4/3) * 3.14159 * r^3
- 375.3148 ft^3 = (4/3) * 3.14159 * r^3
4. To solve for the radius, we can rearrange the formula:
- (4/3) * 3.14159 * r^3 = 375.3148 ft^3
- r^3 = (375.3148 ft^3 * 3) / (4 * 3.14159)
- r^3 = 281.4861 ft^3
- r ≈ ∛(281.4861 ft^3)
- r ≈ 7.0365 ft
Therefore, the radius of the sphere with a volume 40% greater than that of a sphere with a radius of 4 ft is approximately 7.0365 ft.