An asprin tablet in a shape of a right circular cylinder has the height of 2/5 cm and the raidus of 1/2cm. the manufacturer also
wishes to market the asprin in capsule form. The capsule is to be 5/3cm in total length, in the shape of a right cylinder with hemispheres attached to both ends . find a function that represents the volume of the capsule. fund the radius of the capsule so that it is the same volume of the tablet.
Volume of the tablet is height times pi r squared = .4cm x pi x .5^2 = pi cm^3
So, the tablet has a volume of pi cubic centimeters (cc).
We want the capsule to have the same volume. The two hemispherical ends put together make one sphere. The volume of a sphere is 4/3 pi r^3. And the cylindrical part is the same formula as the first one, but we don't know what r is,
height x pi x r^2.
pi cc = height x pie x r^2 + 4/3 pi r^3 Here, we took the value from the original problem and made it equal to the two ends of the capsule (together were the sphere) PLUS the rest (which is a cylinder.)
Now, divide everything by pi to factor it out of the equation.
cc = height x r^2 + 4/3 r^3 The problem told us that the total length is 5/3 cm, this means the cylinder height + the radius times two = 5/3. (Wish I could draw you a picture) So height in the equation at the beginning of this paragraph is 5/3 - 2r.
Now we have volume in cc = (5/3 - 2r)r^2 + 4/3 r^3 =
5/3r^2 - 2r^3 + 4/3r^3
simplified by combining common terms, and written in standard form,
volume in cc = 2 r^3 + 5/3 r^2 = 1 = r^2 ( r - 5/3) , factoring out the r^2.
this means that r^2 is the reciprocal of r - 5/3, or r^2 = 1/(r - 5/3), and this is a quadratic equation.
r^2 - r + .6 = 0 and r^2 -r + .25 = -.6 + .25 by completing the square
or (r - .25)^2 = -.35. Solve for r: square root of r - .25 = square root of .35 or .59
r = .84 cm
