An action movie production team needs glass spheres to hold a green liquid that looks like an explosive. If all of the available 3,392.92 cubic inches of the liquid is to be poured into 30 glass spheres, what should the diameter of each sphere be? Assume that each sphere is filled to the top.
V1 = Volume of one sphere:
V1 = ( 1 / 6 ) * pi * d ^ 3
V = 30 * V1
V = 30 * ( 1 / 6 ) * pi * d ^ 3
V = ( 30 / 6 ) * pi * d ^ 3
V = 5 * pi * d ^ 3
3,392.92 = 5 * pi * d ^ 3
3,392.92 = 5 * 3.1416 * d ^ 3
3,392.92 = 15.708 * d ^ 3 Divide both sides by 15.708
3,392.92 / 15.708 = 15.708 * d ^ 3 / 15.708
215.99949 = d ^ 3
d = third root ( 215.9949 ) = 5,9999952843 in
d = 6 in
well, you know that each sphere holds 3392.92/30 = 113.097
so, now you know that
4/3 pi r^3 = 113.097, so r = 3
To find the diameter of each sphere, we need to first determine the volume of each sphere and then use that volume to calculate the diameter.
Step 1: Find the volume of the liquid.
Since all 3,392.92 cubic inches of the liquid is to be poured into 30 glass spheres, we can divide the total volume by the number of spheres to find the volume of each sphere.
Volume of each sphere = Total volume / Number of spheres
Volume of each sphere = 3,392.92 cubic inches / 30 spheres
Step 2: Find the radius of each sphere.
The volume of a sphere is given by the formula:
Volume of a sphere = (4/3) * π * r^3
To find the radius, we rearrange the formula as follows:
r^3 = (3 * Volume) / (4 * π)
r = ((3 * Volume) / (4 * π))^(1/3)
Step 3: Find the diameter of each sphere.
The diameter of a sphere is simply twice the radius.
Let's calculate the values:
Step 1:
Volume of each sphere = 3,392.92 cubic inches / 30 spheres
Volume of each sphere = 113.09733 cubic inches
Step 2:
r = ((3 * 113.09733) / (4 * π))^(1/3)
r ≈ 2.8397 inches
Step 3:
Diameter of each sphere = 2 * r
Diameter of each sphere ≈ 2 * 2.8397 inches
Diameter of each sphere ≈ 5.6794 inches
Therefore, the diameter of each sphere should be approximately 5.6794 inches.
perimeter
=
2(length + width)
=
2(3 + 3)
=
2(6)
=
12 inches.