A hi-tech bowling ball is constructed from concentric spheres, kind of like a perfectly round M&M peanut. The inner sphere, of material X, has a radius of 1 inch. The middle spherical shell of material Y has a thickness of 2 inches. The outer shell of material Z has a thickness of 1 inch. A ball of the same size made of 100% X,Y, or Z weighs 16, 14, and 12 lbs respectively. What is the weight of this hi-tech ball to the nearest tenth?
The volume of a sphere is
(4π/3)r³
The shell thicknesses are:
X=1", Y=2", Z=1"
The proportions of volumes X, Y and Z are:
X:Y:Z
=(1³-0³):(3³-1³):(4³-3³)
=1:8:55
Therefore weight of ball
=(1*16+8*14+55*12)/(1+8+55)
=788/64
=197/16
=12 5/16 pounds (12 lb 5 oz)
To find the weight of the hi-tech bowling ball, we need to calculate the volume of each material and then determine the weight based on its density.
First, we calculate the volume of each material sphere using the formula for the volume of a sphere: V = (4/3) * π * r^3, where V is the volume and r is the radius.
For material X (inner sphere):
Volume_X = (4/3) * π * (1 inch)^3
For material Y (middle spherical shell):
We need to find the difference between the volumes of the outer and inner sphere to get the volume of the middle shell. The radius of the outer sphere is the sum of the radius of the inner sphere (1 inch) and the thickness of the middle shell (2 inches).
Volume_Y = (4/3) * π * [(1 inch + 2 inches)^3 - (1 inch)^3]
For material Z (outer shell):
We need to find the difference between the volumes of the outermost and middle sphere to get the volume of the outer shell. The radius of the outermost sphere is the sum of the radius of the inner sphere (1 inch), the thickness of the middle shell (2 inches), and the thickness of the outer shell (1 inch).
Volume_Z = (4/3) * π * [(1 inch + 2 inches + 1 inch)^3 - (1 inch + 2 inches)^3]
Now, let's calculate the weights of each material.
For material X:
Weight_X = Volume_X * Density_X
Density_X = 16 lbs per (1 inch)^3
Weight_X = Volume_X * 16 lbs
For material Y:
Weight_Y = Volume_Y * Density_Y
Density_Y = 14 lbs per (1 inch)^3
Weight_Y = Volume_Y * 14 lbs
For material Z:
Weight_Z = Volume_Z * Density_Z
Density_Z = 12 lbs per (1 inch)^3
Weight_Z = Volume_Z * 12 lbs
Finally, we can calculate the weight of the hi-tech ball by summing up the weights of each material:
Weight_hi-tech ball = Weight_X + Weight_Y + Weight_Z
Calculate the volumes and weights for each material and sum them up to find the weight of the hi-tech ball to the nearest tenth.