Trying to find the volume of a bottle using just mathematics

[QueekJJ]

Here is the bottle i chose and plotted into the graph:


I used Desmos to plot this graph, here is the link in case you need a better picture https://www.desmos.com/calculator/ayeq3zjf4j

As you can see, i only know how to plot the red and blue lines, because its straight. However, i dont know what function to use for the rest of the bottle. I considered using y=1/x or y=-1/x, but i dont know if this would work with the formula im going to use

Based on a YouTube video i found, to find the volume i should use the formula bellow:


From what im guessing, the formula above is basically {pi r^2 times legth}, but the r and length is replaced with integration, which then leads me to conclude that integration is finding the area. This then leads to my next question, if the bottle have different shape, what should i do? (For example a rectangle bottle or a weird shape like a ketchup bottle)

 

[Dr.Peterson]

Here is the bottle i chose and plotted into the graph:
View attachment 33372

I used Desmos to plot this graph, here is the link in case you need a better picture https://www.desmos.com/calculator/ayeq3zjf4j

As you can see, i only know how to plot the red and blue lines, because its straight. However, i dont know what function to use for the rest of the bottle. I considered using y=1/x or y=-1/x, but i dont know if this would work with the formula im going to use

Based on a YouTube video i found, to find the volume i should use the formula bellow:
View attachment 33374

From what im guessing, the formula above is basically {pi r^2 times legth}, but the r and length is replaced with integration, which then leads me to conclude that integration is finding the area. This then leads to my next question, if the bottle have different shape, what should i do? (For example a rectangle bottle or a weird shape like a ketchup bottle)

What you're doing here is calculus, not geometry; are we to assume that you haven't learned any calculus other than what you saw in the video? (The integration is not so much "finding the area" as "summing up small changes", in this case volumes of slices). Please tell us what you have learned.

Is this an assignment you were given? If so, I'd like to see the exact wording, to be sure what you were expected to do; and also the context. It might be sufficient to approximate the bottle with a set of cylinders and parts of cones, and use formulas for those. Or, if calculus is appropriate, then approximate the curve with straight lines, parabolas, or other easy curves (not all nice curves are easy to integrate!) and integrate piece by piece.

If it is just a challenge you've given yourself, then the cylinder and cone idea is what I'd do first -- and then compare what I got with the actual labeled or measured volume to see how close I got. You don't get an exact answer no matter what you do, because you won't find an exact function for the shape.

 

[QueekJJ]

What you're doing here is calculus, not geometry; are we to assume that you haven't learned any calculus other than what you saw in the video? (The integration is not so much "finding the area" as "summing up small changes", in this case volumes of slices). Please tell us what you have learned.

Is this an assignment you were given? If so, I'd like to see the exact wording, to be sure what you were expected to do; and also the context. It might be sufficient to approximate the bottle with a set of cylinders and parts of cones, and use formulas for those. Or, if calculus is appropriate, then approximate the curve with straight lines, parabolas, or other easy curves (not all nice curves are easy to integrate!) and integrate piece by piece.

If it is just a challenge you've given yourself, then the cylinder and cone idea is what I'd do first -- and then compare what I got with the actual labeled or measured volume to see how close I got. You don't get an exact answer no matter what you do, because you won't find an exact function for the shape.

Oh sorry i thought it was geometry because of the graph plotting, my bad

Im in pre-u right now, and had used IGCSE as my highschool syllabus. I know how to intergrate but i never understood the meaning behind it, and from that i think you can expect my grade isnt the best.

This was more of an investigation folio where we find the volume of different kinds of bottles, and discuss reasonableness and limitation of the result. Thats basically the entire question.

 

[Steven G]

From x=3 to x=11 you have a cylinder. Find the volume of the cylinder.
From x=11 to x=13 1/2 you have what I would call a negative parabola. Using some points on that parabola find its equation.
Do the same for each different piece of the bottle.
Come back and show us your work.

 

[QueekJJ]

From x=3 to x=11 you have a cylinder. Find the volume of the cylinder.
From x=11 to x=13 1/2 you have what I would call a negative parabola. Using some points on that parabola find its equation.
Do the same for each different piece of the bottle.
Come back and show us your work.

Sorry late reply, was asleep during that time ?


Ohhhhhh for some reason i didnt think it was a cylinder for x=3 to x=11.

The volume of the cylinder (i used 2.5 instead of 3 to make it slightly more accurate to the model):

I was also able to find the volume for the bottle cap part:


Then using the negative parabola for x=0 to x=2.5 , -(0.4x + 0.8)^2 + 1.4x + 2.84, and also for x=11 to x=13.5, -(0.4x - 4.35)^2 + 3.1
Here is a link for a better visual of my attempt to trace the bottle https://www.desmos.com/calculator/fu8cmoe6ng

I used my graphic calculator and got 155.776 for 0 to 2.5 and 144.644 for 11 to 13.5

It was going great and all, but then i got into a new problem. If we add all the result so far, we get 673. That is so much more as the bottle im using is labeled 400ml, and i still havent done x=13.5 to x=20 yet. I even tried using the same formula for the cylinder and i got a abnormally huge number

First picture here is trying the cylinder into the formula, second picture is how i got 155.776 and 144.644

 

[blamocur]

Sorry for not checking your math (getting asleep fast ), but the 400 ml on the label usually refers to the content, not the full volume of the bottle. Moreover, 673 seems to much for the whole bottle too: if the bottle were a 22 cm tall cylinder with radius of 3cm the whole volume would be around 622 cm³, but with such narrow neck it should be noticeably less than even that. If the content is equivalent to a cylinder 14 cm tall (which looks reasonable to me) then its volume would be about close to 400 ml ([imath]\pi \times 3^2 \times 14 \approx 395.84[/imath])

 

[QueekJJ]

Sorry for not checking your math (getting asleep fast ), but the 400 ml on the label usually refers to the content, not the full volume of the bottle. Moreover, 673 seems to much for the whole bottle too: if the bottle were a 22 cm tall cylinder with radius of 3cm the whole volume would be around 622 cm³, but with such narrow neck it should be noticeably less than even that. If the content is equivalent to a cylinder 14 cm tall (which looks reasonable to me) then its volume would be about close to 400 ml ([imath]\pi \times 3^2 \times 14 \approx 395.84[/imath])

Yeh thats what i thought too, but i still cant find the error. I was thinking maybe i input the integration wrong into the calculator, but i triple checked and retype it again, but still same answer

The next thing i could think off is that the scale of the bottle is not accurate. I do have an exact same bottle with me, as far as i can measure with my ruler, the height is 26.5-27 cm and the width is 6-6.2cm. It was hard to measure with just 2 pairs of 15cm rulers but im pretty sure its somewhat accurate. This means that the scale is not the problem

 

[blamocur]

Yeh thats what i thought too, but i still cant find the error. I was thinking maybe i input the integration wrong into the calculator, but i triple checked and retype it again, but still same answer

The next thing i could think off is that the scale of the bottle is not accurate. I do have an exact same bottle with me, as far as i can measure with my ruler, the height is 26.5-27 cm and the width is 6-6.2cm. It was hard to measure with just 2 pairs of 15cm rulers but im pretty sure its somewhat accurate. This means that the scale is not the problem

I find it very difficult to understand, and thus verify, different parts of your computations. If you make a neat summary of all parts, listing the volume and the description of the part (i.e., X range) it might be easier to spot the error. I've noticed that when I try to make things more readable/presentable I often end up finding errors myself.