Interesting Maths Problem!

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MathsKid007

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A lightweight "pop-up" tent consists of six plastic struts that are inserted into pockets sewn into the joins of the fabric panels. The resulting shape has hexagonal cross-sections, while vertical cross-sections through the center are semicircular. The overall height is (unique number / 55) meters.

In order to find whether it is dafe to use a camping gas lamp inside, a camper needs to know the volume of air in the tent.

a. By using Riemann's sum, synthesise a mathematical model for finding the exact volume of any lightweight "pop-up" tent of height "h".


Unique Number is 104. Goodluck
 
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MathsKid007 said:
View attachment 9719Unique Number is 104. Goodluck :)
Click to expand...
Please share your work/attempts.

attachment.php

What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for
 
MathsKid007 said:
A lightweight "pop-up" tent consists of six plastic struts that are inserted into pockets sewn into the joins of the fabric panels. The resulting shape has hexagonal cross-sections, while vertical cross-sections through the center are semicircular. The overall height is (unique number / 55) meters.

In order to find whether it is dafe to use a camping gas lamp inside, a camper needs to know the volume of air in the tent.

a. By using Riemann's sum, synthesise a mathematical model for finding the exact volume of any lightweight "pop-up" tent of height "h".


Unique Number is 104. Goodluck :)
Click to expand...
Why would we need the luck? We're not the one with the homework exercise. I see that your instructor has found a way to cause different students (or groups, if this is a group-project exercise) to arrive at different numerical values, but the "unique number" you were assigned is not necessary for part (a).

Did you do anything based on the very similar question posted on this site in 2013? Or this post by the same person but to another site? Did the placement of the question (on page 8 of this document), immediately following a pyramid question, suggest anything to you?

If you are still having difficulty in answering this, please reply with a clear listing of your thoughts and efforts so far. Thank you! ;)
 
What does "the overall height is Unique Number divided by 55" mean?

A cross-section parallel to the base of either tent is a hexagon. A cross-section through one of the struts is a semi-circle. Taking the height to be "h", we can write \(\displaystyle x^2+ y^2= h^2\) so that, for each y, \(\displaystyle x= \sqrt{h^2- y^2}\) and that will be a diagonal in the hexagon. The hexagon can be divided into 6 equilateral triangles of side length \(\displaystyle \frac{x}{2}= \frac{1}{2}\sqrt{h^2- y^2}\) or into 12 right triangles with hypotenuse of that length, base of length \(\displaystyle \frac{1}{4}\sqrt{h^2- y^2}\) and height \(\displaystyle \sqrt{3}{4}\sqrt{h^2- y^2}\). So the area of the hexagon is "12 times 1/2 base times height" or \(\displaystyle (12)\left(\frac{1}{2}\right)\left(\sqrt{3}{4}\sqrt{h^2- y^2}\right)\left(\frac{1}{4}\sqrt{h^2- y^2}\right)= \frac{3}{8}\sqrt{3}(h^2- y^2)\). Multiplying that by the infinitesimal thickness, dy, we have \(\displaystyle \frac{3}{8}\sqrt{3}(h^2- y^2)dx\). Integrating that gives the volume of the tent, \(\displaystyle \int_0^h \frac{3}{8}\sqrt{3}(h^2- y^2)dx\).

The second tent also has hexagonal cross-sections but now the struts are straight lines. Taking the base to have sides of length "a" as in your picture, the semi-diagonal at the base will be a but decrease linearly to 0 at height y= h. We can write this as d(y)= Ay+ B. d(0)= B= a and d(h)= Ah+ a= 0. A= -a/h. The semi-diagonal at height h is a- (a/h)y= a(1- y/h). Again, taking the infinitesimal of thickness of a hexagon to be dy, the differential of volume is a(1- y/h)dy. The volume of the tent is \(\displaystyle a\int_0^h (1- y/h)dy\).
 
Volume of Hexagonal Pyramid - Using Integration from (0-h)

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Vinculum said:
The Question:
By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h.
Diagram:
View attachment 9825
What I Have:

View attachment 9826
Where do I go from here?
Note, I must integrate from 0-h.
Click to expand...

This is the "easy" part (just algebra). You know that s(y) is a linear function such that s(0) = S and s(h) = 0. (I've used different cases for the function (side at height h) and the constant (side of base), which is important to avoid confusion.)

So what is the equation of the straight line through the points (0,S) and (h,0)? As you said, use the point-slope formula -- or, if you prefer, the slope-intercept form, which is simpler.

Then put that function into the integral, and integrate.

Let's see your work. Everything has been fine so far.
 
This user posted this problem on another site a few days ago, and I gave a complete solution. They then took part of my work and went to another site and asked for help, where I also gave a complete solution. Now, they have taken a bit more of my work and come here. I don't get it, but I'm not going to post my work a third time.
 
The path this exercise has taken in our forum (since July 8th) reminds me of the Single Bullet Theory, heh.

It started with MathsKid007, posting a growing sequence of work, in a thread that began as almost a challenge to us titled "Interesting Maths Problem!" (Originally, MathsKid007 wished us luck. Posts were then edited, a few times.) The thread was eventually moved off the Math Odds & Ends board; ten days later, Vinculum started a similar thread. The two threads were combined. Subhotosh and Stapel each asked to see work. Work was subsequently edited in, over stages. Halls provided explanation, ending with an integral for the volume.

(I don't know whether MathsKid007 and Vinculum are the same individual, but they share the same IP address. Same math camp?)

Now, another ten days have passed, and we're back. I expect this thread to be merged into the other (combined) thread, soon. :cool:
 
mmm4444bot said:
The path this exercise has taken in our forum (since July 8th) reminds me of the Single Bullet Theory, heh.

It started with MathsKid007, posting a growing sequence of work, in a thread that began as almost a challenge to us titled "Interesting Maths Problem!" (Originally, MathsKid007 wished us luck. Posts were then edited, a few times.) The thread was eventually moved off the Math Odds & Ends board; ten days later, Vinculum started a similar thread. The two threads were combined. Subhotosh and Stapel each asked to see work. Work was subsequently edited in, over stages. Halls provided explanation, ending with an integral for the volume.

(I don't know whether MathsKid007 and Vinculum are the same individual, but they share the same IP address. Same math camp?)

Now, another ten days have passed, and we're back. I expect this thread to be merged into the other (combined) thread, soon. :cool:
Click to expand...

At MHB and MHF, this question was posted under the username MathsKid007. At MMF, the "domed" version has been posted by MathsKid007, but not the pyramidal version...yet. I get that impatient students who have procrastinated will throw a wide net to maximize their chances of getting a quick solution, but I don't get why they would post at one site, and get a complete solution, then take part of that solution and post elsewhere. Why not ask for clarification if something doesn't make sense, of the person who replied to their first posting of the problem. But, I'm not complaining really, just musing. :)
 
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