The smaller cannonball weighs approximately 89% less than the larger cannonball.
What are the steps to the solution above?
Step 1 - Identify the Formula for the Bigger Spherical Cannonball.
Recall that the volume of a sphere is given as V = (4/3)π(db/2)³........1
Expanding the bracket, we have V = (4/3)πdb³/8
Divide all though by 8 and the result is
⇒ Vb = (1/6)πdb³..........2
Recall that we know that mass is given as Density multiplied by Volume; that is
m = DV
Therefore,
m = (1/6)πdb³*463..........3
Step 2 - Let us assume that the mass of the bigger sphere is 1 pound. So we set equation 3 to 1.
1 = (1/6)πdb³*463.
Work to make db the subject of the formula; and we have
⇒ db³ = 6/463*π
⇒ db = ∛6/(463*π)
⇒ db = 0.160 ft.
From the question, we know that the diameter of the smaller sphere is 1 inch less than the diameter of the larger cannonball.
Hence,
1 inch = 0.0833 foot
⇒ ds = db - 0.0833
⇒ ds = 0.160 - 0.0833
⇒ ds = 0.077 ft
Step 3 - to find the volume of the smaller sphere we state:
Vs = (4/3)π(ds/2)³
⇒ Vs = (4/3)π(0.077/2)³
⇒ Vs = 0.000239
Recall that mass of a sphere is m = DV. Therefore, the mass of the smaller cannonball is:
m = 463*0.000239
⇒ m = 0.1106 lb
Step 4 - find the percentage difference in the mass of both spheres:
Percentage Difference in Mass of both sphere = (1 - 0.1106/1)*100
Hence the difference = 88.94% which is approximately 89%.
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