Volume of pyramid formed by center of 5 spheres inside a hemisphere

[songoku]

Homework Statement

Please see below

Relevant Equations

Volume of hemisphere = $\frac{4}{3} \pi r^3$

Volume of pyramid = 1/3 x base area x height

Let the radius of the small sphere = r

3r = 1 → r = 1/3

$x=\sqrt{4r^2-r^2}=r\sqrt{3}$

Volume of pyramid:
$$=\frac{1}{3} \times (2r\sqrt{3})^2 \times r$$
$$=\frac{4}{27}$$

So m + n = 31, but the answer is 29.

I guess my mistake is assuming line AB is tangent to the top sphere. How to do this question?

Thanks

 

[haruspex]

Homework Statement:: Please see below
Relevant Equations:: Volume of hemisphere = $\frac{4}{3} \pi r^3$

Volume of pyramid = 1/3 x base area x height

View attachment 301746

Let the radius of the small sphere = r
View attachment 301748

3r = 1 → r = 1/3

$x=\sqrt{4r^2-r^2}=r\sqrt{3}$

Volume of pyramid:
$$=\frac{1}{3} \times (2r\sqrt{3})^2 \times r$$
$$=\frac{4}{27}$$

So m + n = 31, but the answer is 29.

I guess my mistake is assuming line AB is tangent to the top sphere. How to do this question?

Thanks

You should not assume that, you should prove it. Which is easy.
$2r\sqrt{3}$ is the length of what part of the pyramid?

 

[songoku]

You should not assume that, you should prove it. Which is easy.
$2r\sqrt{3}$ is the length of what part of the pyramid?

$2r\sqrt{3}$ is the length of line AB, which is the one side of base of the pyramid.

Sorry I don't know how to prove it. From the picture on the left, it seems AB is not the tangent to the top sphere.

 

[haruspex]

$2r\sqrt{3}$ is the length of line AB, which is the one side of base of the pyramid.

Sorry I don't know how to prove it. From the picture on the left, it seems AB is not the tangent to the top sphere.
View attachment 301768

You seem to be confused about the arrangement. Look at the three spheres shown in the right hand diagram in the problem statement. Which three do they correspond to in the left hand diagram?

 

[songoku]

You seem to be confused about the arrangement. Look at the three spheres shown in the right hand diagram in the problem statement. Which three do they correspond to in the left hand diagram?

 

[haruspex]

View attachment 301769

Compare how close 1 and 3 are to each other in those two diagrams. Do they really look the same?

 

[Lnewqban]

View attachment 301769

 

[songoku]

Compare how close 1 and 3 are to each other in those two diagrams. Do they really look the same?

Ah I understand and I got the answer. I am also able to prove that line AB is tangent to the top sphere but I did it by using equation of circle and straight line. Is there easier way to prove that it? Maybe using geometry or something else?

Thanks

 

[haruspex]

I am also able to prove that line AB is tangent to the top sphere but I did it by using equation of circle and straight line. Is there easier way to prove that it? Maybe using geometry or something else?

Thanks

Reflect the elevation (cross-section) diagram about the horizontal plane to get six spheres in a larger sphere. Do you see the hexagon and its equilateral triangles?

Btw, the plan view provided with the question is a bit misleading. It should not show the spheres touching the enclosing circle if that is supposed to represent a circumference of the hemisphere. As you can see from the elevation view, there should be a small gap.

 

[Lnewqban]

What is the meaning of (m, n) = 1?

 

[songoku]

Reflect the elevation (cross-section) diagram about the horizontal plane to get six spheres in a larger sphere. Do you see the hexagon and its equilateral triangles?

Yes, I see it

What is the meaning of (m, n) = 1?

I am not so sure but my guess is HCF of m and n is 1

Thank you very much haruspex and Lnewqban

 

[haruspex]

my guess is HCF of m and n is 1

Yes, it's one way to say that m and n are coprime.

 

[Prof B]

What is the meaning of (m, n) = 1?

(m,n) stands for the greatest common divisor of m and n.

 

[Lnewqban]

Thanks to @songoku, @haruspex and @Prof B.

Now, I don't understand how the volume of the pyramid, m/n, as well as m+n are related to the radius of the semisphere, to the coprime fact and to prove that line AB is tangent to the top sphere.

Could you songoku show your revised work that led you to the correct answer, please?

 

[songoku]

Thanks to @songoku, @haruspex and @Prof B.

Now, I don't understand how the volume of the pyramid, m/n, as well as m+n are related to the radius of the semisphere, to the coprime fact and to prove that line AB is tangent to the top sphere.

Could you songoku show your revised work that led you to the correct answer, please?

View attachment 301770

Let the radius of the small sphere = r
View attachment 301748

3r = 1 → r = 1/3

$x=\sqrt{4r^2-r^2}=r\sqrt{3}$

Based on your diagram and what I did in post#1, line AB is the diagonal of the base of the pyramid. Let $a$ is the length of the base of the pyramid, then:
$$(AB)^{2}=2a^2$$
$$(2x)^{2}=2a^2$$
$$12r^2=2a^2$$
$$a^2=6r^2$$
$$a^2=6\left(\frac{1}{3}\right)^{2}$$
$$a^2=\frac{2}{3}$$

Volume of pyramid:
$$=\frac{1}{3}a^2r$$
$$=\frac{1}{3} . \frac{2}{3}.\frac{1}{3}$$
$$=\frac{2}{27}$$

Since 2 and 27 are coprime, so m = 2 and n = 27 then m + n = 29

 

[Lnewqban]

Thank you very much for your response, @songoku ; it is much clearer to me now.

I have drawn section A-A at scale, showing the main dimensions, the hexagon mentioned by @haruspex , as well as the 0.08 gap that should be seen between the edge of the semi-sphere glass cover and the (4) smaller glass spheres when looking from above (elevation view).

 

[songoku]

Thank you very much for the help and explanation haruspex, Lnewqban, Prof B

 

[Prof B]

Now, I don't understand how ... to prove that line AB is tangent to the top sphere.

Draw the hexagon as recommended by haruspex. Each of its internal angles is 120 degrees. Three consecutive vertices of that hexagon form an isosceles triangle AXB, so its other angles are 30 degrees.
Let M be the midpoint of AB. Two of the angles in MBX are 30 and 60 degrees. Since the angles in a triangle add up to 180 degrees, the other angle must be 90 degrees, i.e., the angle XMB is a 90 degree angle. XM is a radius of the top circle. The tangent to that circle at M is perpendicular to the radius XM. BM is also perpendicular to XM. Since only one line through M is perpendicular to XM, BM must be tangent to the top circle.

Note: The base of an isosceles triangle that has sides of length $s$ and top angle $\theta$ is equal to $2s\sin(\theta/2)$, which in our problem is $2 \cdot 2r \cdot \sqrt{3}/2 = 2r\sqrt{3}$, so the fact that AB is tangent to the top circle is merely tangential to the solution of the problem.