three spheres of diameter 2,3 & 4cms formed into a single sphere.Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of diameter.
volume of the original spheres
= (4/3)π (1^3 + (3/2)^3 + 2^3)
= (4/3)(99/8)π or 16.5π
radius of big sphere ---> r cm
(4/3)π r^3 = (4/3)(99/8) π
r^3 = 99/8
r = 99^(1/3)/2 or the cuberoot of 99 /2
diameter = 2r = cuberoot(99) = appr 4.626 cm
To find the diameter of the new sphere, we need to apply the given information that the volume of a sphere is proportional to the cube of its diameter.
Let's first find the volumes of the three spheres:
- Sphere 1: Diameter = 2 cm
Volume = (4/3) * pi * (2/2)^3 = (4/3) * pi * 1^3 = (4/3) * pi = V1
- Sphere 2: Diameter = 3 cm
Volume = (4/3) * pi * (3/2)^3 = (4/3) * pi * (27/8) = 9 * pi / 2 = V2
- Sphere 3: Diameter = 4 cm
Volume = (4/3) * pi * (4/2)^3 = (4/3) * pi * 2^3 = (4/3) * pi * 8 = 32 * pi / 3 = V3
Now, let's add up the volumes of the three spheres to find the total volume:
Total Volume = V1 + V2 + V3
Next, we need to find the diameter of the new sphere. Since the volume of a sphere is proportional to the cube of its diameter, we can set up the following equation:
Total Volume = (4/3) * pi * (new diameter/2)^3
Equating the two expressions for total volume, we have:
V1 + V2 + V3 = (4/3) * pi * (new diameter/2)^3
Simplifying the equation, we get:
V1 + V2 + V3 = (4/3) * pi * (new diameter/8)
To find the new diameter, we can multiply both sides of the equation by 3/4pi:
(new diameter/8) = (3/4pi) * (V1 + V2 + V3)
Now, multiply both sides by 8 to isolate the new diameter:
new diameter = 8 * (3/4pi) * (V1 + V2 + V3)
Substituting the values of V1, V2, and V3, we get:
new diameter = 8 * (3/4pi) * [(4/3) * pi + 9 * pi / 2 + 32 * pi / 3]
Simplifying the expression inside the brackets:
new diameter = 8 * (3/4pi) * [(4/3) * pi + 27 * pi / 2 + 32 * pi / 3]
Combining the terms with pi:
new diameter = 8 * (3/4pi) * (8/3pi + 81/2pi + 32pi/3)
new diameter = 8 * (3/4) * (8/3 + 81/2 + 32/3)
Simplifying the expression inside the brackets:
new diameter = 8 * (3/4) * (16/6 + 324/6 + 64/6)
new diameter = 8 * (3/4) * (404/6)
new diameter = 8 * (3/2) * (202/6)
Simplifying further:
new diameter = 4 * (3) * (101/3)
new diameter = 12 * (101/3)
Finally, we have:
new diameter = 404/3
Therefore, the diameter of the new sphere, formed by combining the three given spheres, is 404/3 cm.
To find the diameter of the new sphere formed by combining the three spheres, we need to consider the volumes of the individual spheres.
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 diameter (D) is twice the radius (r), we can rewrite the formula as V = (4/3) * π * (D/2)^3.
Now, let's calculate the volumes of the three spheres:
Sphere 1: Diameter = 2 cm
Radius = Diameter / 2 = 2 / 2 = 1 cm
Volume = (4/3) * π * (1)^3 = (4/3) * π * 1 = 4/3π
Sphere 2: Diameter = 3 cm
Radius = Diameter / 2 = 3 / 2 = 1.5 cm
Volume = (4/3) * π * (1.5)^3 = (4/3) * π * 3.375 = 4.5π
Sphere 3: Diameter = 4 cm
Radius = Diameter / 2 = 4 / 2 = 2 cm
Volume = (4/3) * π * (2)^3 = (4/3) * π * 8 = 32/3π
To find the volume of the new sphere formed by combining the three spheres, we add up their volumes:
Total Volume = (4/3π) + (4.5π) + (32/3π) = (4/3 + 4.5 + 32/3) * π = (8/3 + 4.5 + 32/3) * π = (8 + 13.5 + 32/3) * π = (40 + 32/3) * π = (120/3 + 32/3) * π = (152/3) * π.
Now, to find the diameter (D_new) of the new sphere, we need to find the radius (r_new) using the volume formula and rearranging the equation:
V = (4/3) * π * r_new^3
(152/3) * π = (4/3) * π * r_new^3
(152/3) = (4/3) * r_new^3
r_new^3 = (152/3) * (3/4)
r_new^3 = 152/4
r_new^3 = 38
r_new = ∛38
Since D_new = 2 * r_new, the diameter of the new sphere is:
D_new = 2 * ∛38 ≈ 7.621 cm.
Therefore, the diameter of the new sphere formed by combining the three spheres is approximately 7.621 cm.