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When a frog swims under water, it can only absorb new oxygen through its skin. For example, bullfrogs absorbs oxygen through their skin with a rate constant $k_1= 11 \times 10^{-5} \text{g O}_2 \text{cm}^{-2}\text{hr}^{-1}$ and consume oxygen at a rate proportional to their volume (every cell consumes O$_2$) with a rate constant $k_2=3.8 \times 10^{-5} \text{g O}_2\text{ cm}^{-3}\text{hr}^{-1}$.

While the shape of a frog is not simple (i.e. not just a sphere or cylinder), its volume and surface area can be related to its length ($L$) from head to tail by a simple relationship of the form

$V(L)=pL^3 $

where $p$ and $q$ are constants for a given species. For bullfrogs, $p=0.3$ and $q=4.2$.

Here we ask you to investigate how frogs of various sizes can tolerate long periods under water.

  1. Before doing any calculations, describe in a couple sentences what you expect to find for the relationship of frog size to longest time under water, e.g. frogs larger than some threshold size can stay under water longer or a frog can stay under for a period of time proportional to its size etc. You will be marked not for the correctness of your answer but for its clarity. This is to encourage you to establish an expectation before doing calculations.
  2. At what size $L_{bal}$ do a bullfrog's absorption and consumption rates balance?
  3. At what size $L_{bal}$ do a frog's absorption and consumption rates balance? Note that this is not asking about the specific case of bullfrogs so you should leave $k_1, k_2, p$ and $q$ as unspecified constants.
  4. Describe using words and inequalities what this model predicts will happen to the amount of oxygen in a frog's body if the frog is (a) larger and (b) smaller than the length $L_{bal}$. Use a hand-drawn graph of consumption and absorption as functions of length to illustrate. Label the consumption and absorption functions $C(L)$ and $A(L)$ and the location of $L_{bal}$ on the $L$ axis. Again, this is not asking about the specific case of bullfrogs so you should leave $k_1, k_2, p$ and $q$ as unspecified constants.

To think about (not to hand in):

  • Why (or why not) does your answer to 4 differ from your answer to 1?
  • When absorption is greater than consumption, this model predicts an ever increasing amount of oxygen in the frog. This is not realistic. What modification to the model will make it more realistic with regard to this prediction?

See the related example if you are not sure what is expected of you in terms of "good mathematical communication".