3D geometry – A plant in your stomach (basic)

Concepts addressed: percentages, date calculation, sphere volume.
Recommended grade: 8th.
Difficulty level: Basic.
An advanced version of A plant in your stomach is also available.


Remember how, as a kid, we were told that if we accidentally swallowed a fruit seed, a tree will grow inside our stomachs? It seems funny today, but oh man, we didn’t feel like laughing back then. Fortunately, today we know that such a thing is impossible, but what if it wasn’t?

  1. Let’s say that you swallowed a watermelon seed. Assume for simplicity that your stomach is in the shape of a sphere with a diameter of seven inches, and that watermelons also grow to be spherical. What is the maximum volume that a watermelon could possibly reach inside your stomach, if your stomach’s muscles can stretch so that it’s volume is 10 % larger?
  1. Say that a watermelon has actually grown in your stomach to the size from the first question, and people start asking when you are due. Assuming that a cubic inch of watermelon weighs 1.1 ounces, and your stomach digests 12 ounces every day, when will you be fully rid of the nasty fruit (counting from today’s date)? For simplicity, we assume that you don’t eat anything else during those days, that the watermelon stopped growing altogether once it reached the limit, and that you only started to digest it once it reached it maximum size.

Useful calculators:

Question 1 hints:

Hint 1
Try to sketch the shape of the stomach. Can you describe its dimensions?
Hint 2
What is the volume of your stomach before its muscles stretch to their limits? And after?

Question 2 hints:

Hint 1
From the first question, we know the volume of the watermelon, but how much does it weigh?
Hint 2
Knowing the mass of our watermelon, how many days will you need to digest all of it?
Hint 3
What will be the date you digest the last bit? Remember that you digest the first 12 ounces today.

Solutions (WARNING: depend on the current date, example for 30th March 2020):

Question 1
V ≈ 197.5 in³
Question 2
On 17th April.

Step-by-step solution:

Question 1
If the stomach has a diameter of 7 in, then its radius is half that, which is r = 3.5 in. Using this information, we can calculate the volume of the stomach before the muscles stretch with the sphere calculator:
V = (4 / 3) * π * r³ ≈ 179.5 in³.

We know that muscle stretching increases this volume by 10%. Therefore, it will make the stomach’s volume to be 100% + 10% = 110% of what we calculated above. The percentage calculator gives that this is
V_max = 110% * V 197.5 in³,
and that is the maximal volume our watermelon can reach.

Question 2
From the first question, we got that the volume is 197.5 in³. If we now use the information given in the scenario, we can count that it will weight
197.5 * 1.1 = 217.25 oz.
Now, if every day we digest 12 oz, then we will need
217.25 / 12 18.1
days to be rid of the watermelon, so we’ll need 19 days (including today). Therefore, if we start digesting on 30th March 2020, then according to the date calculator, we’ll finish on 17th April 2020.
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