Question

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.

Answer 1

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