Recent content by bob012345

I like old stuff from the 60’s like ‘Eight Miles High’ by the Byrds. Commercial jets usually fly six to seven miles high but on a recent flight we got very close to 8 miles at 41,000 ft. to get above some choppy air.

Here is my proof. $$r_1^2+(r_1+d)^2=R^2=r_2^2+(r_2-d)^2$$ Solving for ##d##$$d=\frac{r_2^2-r_1^2}{r_2+r_1}=r_2-r_1$$ Therefore $$r_1^2+r_2^2=R^2$$ And area ratio is $$\frac{\frac{1}{2}\pi r_1^2+\frac{1}{2}\pi r_2^2}{\pi R^2}=\frac{1}{2}$$

No, that is not my proof, just an observation.

The extreme cases I mentioned are when the radius of the smaller semi-circle goes to zero, the larger semi-circle fills exactly half the circle and also when the radii of the semi-circles are the same length.

Here is a puzzle from Catriona Shearer. Find the area ratio of the shaded semi-circles to the circle which they fit in. The bases of the semi-circles are parallel. Hint: consider the extreme cases to figure what the answer might be then prove the general case.

Also, the minimum age for Ivy such that Holly is alive 15 years ago is 19.5 years old.

We don’t have enough information to calculate the ages of Holly and Ivy but we can determine the relationship between them. Using the original conditions which we calculated at 6 years ago, we can say that ##H=2I-24## so Holly’s age is always twice that of Ivy minus 24 years. ##H## and ##I##...

I think there is a bit more to explore to this problem. For example, what is the current relationship between the ages of Holly and Ivy? What can we say about who is older or younger? Also, what constraints can we place on the minimum ages if we demand both were alive at the different times...

That is until someone falls into a black hole and gets spaghettified.

At the time I couldn’t figure out how to make Latex to do ##\approx##. Now I do and I edited the above post. Thanks. Regarding ##d=\sqrt{3}r##, the vector from the origin to the center of the small sphere has x,y,z components each equal to ##r##. Regarding the graph, it just visualizes the...

Looking at a projection of the 3D case we can see how the ball is smaller for the same ##R##.

Here is the next part; In the case of 4D, ##d=\sqrt{4}r=2r## then ##r=\frac{R}{3}##. In the general case ##d=\sqrt{n}r## and the ratio $$\frac{r}{R}=\frac{1}{\sqrt{n}+1}$$ The general equation for hypersphere volume is $$V_n(R) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2} + 1\right)} \...

Here is the first part; Let the center to center distance be ##d## and ##r## the smaller circle radius and ##R## the larger circle radius so ##d+r=R##. We know ##d^2=r^2+r^2## or ##d=\sqrt{2}r##. Solving for ##r## in terms of ##R## we get $$r=\frac{R}{\sqrt{2}+1}$$ therefore the fraction that...

Here is an interesting geometry problem given for general interest. In the figure, compute the fraction that is shaded. Then extend that to a sphere snugly packed in an octant of a larger sphere. That means the smaller sphere is tangent to each plane and to the larger sphere in the most compact...

I don’t know if this is simpler but it can be done as the intersection of a line and a circle. Note, I’m using ##x## as the standard coordinate. Referring to @kuruman ’s excellent graphic, If the origin is at A and letting D be along the x axis, the point C is the intersection of the line...