3.A drug manufacturing company wants to manufacture a capsule that contains a spherical pill inside. The diameter of the pill is 4mm and the capsule is cylindrical, with hemispheres on either end. The length of the capsule between the two hemispheres is 10mm. Describe how we could find the exact volume the capsule will hold, excluding the volume of the pill. What is the value that you get from your calculation? Why is it important for us to be able to determine the exact volume of that capsule?
2007-01-31 11:39:28 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
The hemispheres on the end, taken together, equal the volume of a sphere of equal diameter. The radius of the pill is 4/2 = 2. The volume of the sphere is 4/3 * pi * r^3 = 32pi/3.
The "cylinder" between the hemispheres is 10 mm long. The cross-section of the cylinder is a circle, the area of which is pi*r^2 = 4pi. The volume of the cylinder is 10*4pi = 40pi.
The total volume of the cylinder is the sum of the three parts, or 40pi + 32pi/3 = 152pi/3.
Excluding the volume of the pill (which is equal to the two hemispheres), the remaining volume is that of the cylinder, or 40pi.
Why it's important to know that is anybody's guess. Good luck with that one.
Hope that helps.
2007-01-31 11:50:34 · answer #1 · answered by Tim P. 5 · 1⤊ 0⤋
So the capsule is a cylinder with a hemisphere on each end. This means the radius of the hemisphere must equal the raidus of the cylinder (middle) part.
Assuming the spherical pill fits snugly in the cylinder (which we're not told, but there's no way to solve this otherwise), then the diameter of the cylinder is equal to the diameter of the sphere. And since the diameter of the cylinder must match the diameters of the hemispheres, then the volume of the two hemispheres together is the volume of the pill inside. But since the total volume of the capsule is the cylinder part plus the two hemispheres, this means that the remaining volume with the pill inside is just the volume of the cylinder part itself.
The volume of a cylinder is pi*(r^2)h. The radius of the cylinder is 4mm/2 = 2mm. The length of the capsule "between the two hemispheres is 10mm", so that's the length of the cylinder. Putting it all together, you get pi(2^2)10 = 40pi square milimeters.
Why is it important for us to be able to determine the exact volume of that capsule? I guess so that we can manufacture a pill that's large enough to hold what's inside but small enough to swallow?
2007-01-31 20:02:53 · answer #2 · answered by Anonymous · 0⤊ 0⤋
Iassume the diameter of the cylinder is equal to the diameter of the sperical pill and that the total length of the pill is 10 mm.
Then volume of capsule = volume of cylinder + 2*volume of hemishphere = volume of pill
The volume of the pill is 4/3 pir^3= 32pi/3 mm^3 so the voume of the 2 hemispheres is 32pi/3.
The volume of the cylinder is pir^h. h= 10-2r = 10 -4 = 6
So volume of cylinder = pi*2^2*6 = 24pi mm^3
Volume of capsule = 24pi + 32pi/3 mm^3
The volume of the capsule - the volume of the pill is 24pi mm^3.
I don't know. It seems to be 2 mm longer than it needs to be.
2007-01-31 20:07:31 · answer #3 · answered by ironduke8159 7 · 0⤊ 0⤋