A company is deciding whether to package a ball in a cubic in a cubic box or a cylinder box. In either case, the ball will touch the bottom, top, and sides. What portion of the space inside the cylindrical box is empty?
+1038 This site calculator will solve symbolic math. You can use it to test your solutions.
This returns the solution for the question above.
solve→c1,d,c2,c3(c1=π×d36c2=π×d34c3=(1−(c1c2)))⇒{c1=0d=0c2=0c3=r12c1=r133×π6d=r13c2=r133×π4c3=13}
The result is the same as CPhill’s solution.
Note that all the Cs & Ds are not affecting the result. This calculator has a high resistance to CDD
Here’s another one
solve→c1,d,c2,x(c1=2×π×0.5dc2=2×π×(0.5d+x)c2=c1+1)⇒{c1=r9×πd=r9c2=r9×π+1x=1(2×π)}
^ - - - Does anyone know what it is for? It’s easy.
This one is totally symbolic.
solve→c2(c2×d2t2=c1×d1t1)⇒c2=c1×d1×t2(d2×t1)
+130474 The volume of the cylinder is
V = pi * r^2 * h where r is the radius of the cylinder and h is the height.
And the radius of the ball = the radius of the cylinder.
But since the ball touches the sides, top and bottom, the height of the cylinder must be 2r.
So the volume of the cylinder is
V = pi * r^2 * 2r = 2pi *r^3
And the volume of the ball is
V = (4/3)pi* r^3
So..... the portion of the cylinder that is filled by the ball (in terms of a ratio) is
[(4/3)pi * r^3] / [ 2 pi * r^3] = (4/3)/2 = 4/6 = 2/3 of the cylinder
So....the empty portion, in terms of ratio = 1/3 of the cylinder
+1038 This site calculator will solve symbolic math. You can use it to test your solutions.
This returns the solution for the question above.
solve→c1,d,c2,c3(c1=π×d36c2=π×d34c3=(1−(c1c2)))⇒{c1=0d=0c2=0c3=r12c1=r133×π6d=r13c2=r133×π4c3=13}
The result is the same as CPhill’s solution.
Note that all the Cs & Ds are not affecting the result. This calculator has a high resistance to CDD
Here’s another one
solve→c1,d,c2,x(c1=2×π×0.5dc2=2×π×(0.5d+x)c2=c1+1)⇒{c1=r9×πd=r9c2=r9×π+1x=1(2×π)}
^ - - - Does anyone know what it is for? It’s easy.
This one is totally symbolic.
solve→c2(c2×d2t2=c1×d1t1)⇒c2=c1×d1×t2(d2×t1)
