Answer :
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
