Question

http://prntscr.com/95dtao

Answer 1

@pooja195

Answer 2

I can help you with this. :D
First, we will need to list the two formulas:
The formula for a sphere and the formula for a cylinder. This will help you picture why:

http://docs.unity3d.com/uploads/Main/CapsuleColliderDiagram.svg

\[V=\frac{ 4 }{ 3 }(\pi)(r ^{3}) + (\pi)(r ^{2})(h)\]
The capsule is 12 mm long. This is equal to the height of the cylinder minus the circumference of the sphere (or the radius of each ''half-sphere''. The website I put above can help you understand why this works. If not, please message me. :D
\[h+2r=12\]
Solving for ''h'', we get:
\[h=12 - 2r\]
We can now plug this in for 'h'' for our equation above.
\[V=\frac{ 4 }{ 3 }(\pi)(r ^{3}) + (\pi)(r ^{2})(12-2r)\]
Now, you can distribute the ''12-2r'' to 2 terms before it:
\[V=\frac{ 4 }{ 3 }(\pi)(r ^{3}) + 12(\pi)(r ^{2}) - 2(\pi)(r ^{3})\]
The first and last term have like terms. As such, we can simplify the equation to:
\[V=\frac{ -2 }{ 3 }(\pi)(r ^{3}) + 12(\pi)(r ^{2})\]
Now, the only variable left is the radius (''r''), which we are looking for. From here on, we can plug in the answer choices into our equation.
First off, since this is a ''real-world'' situation, A can be eliminated. In real life, you CANNOT have a negative radius of a capsule.
We are given that the volume (V) is 250 cubic millimeters. There MUST be a radius that obtains this number. If there was no radius, you couldn't find the volume (it would not be possible to have one). Therefore, answer choice D is eliminated.
We are left with B and C. Let's plug in B into our equation:
\[V=\frac{ -2 }{ 3 }(\pi)(2.8^{3}) + 12(\pi)(2.8^{2})\]
Let's solve it:
\[V=\frac{ -2 }{ 3}(\pi)(21.952) + 12(\pi)(7.84)\]\[V=\frac{ -2 }{ 3 }(68.964241) + 12(24.630086)\]
\[V=(-45.976161) + (295.561036)\]
\[V \approx 250\]
This proves that answer choice B, approximately 2.8 millimeters, is correct. If we plug in the other answer choice, C, we would get a volume that is much bigger. Therefore, C would not have worked. B is the correct answer.
If you have any questions, please let me know by message or tagging me. :D