OSH/1/Solution

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These are the marking instructions given to the markers. It is both a solution and a guide to the allocation of points.

Solution key

A suitable solution should look something like this:

  1. A clearly stated prediction should go here. For example: I predict that a larger frog will be able to spend more time underwater, because a larger frog can store more oxygen in its body.
  2. Using the information provided, the a bullfrog of length $L$ will abosorb oxygen at a rate of
    $A=k_1S(L) = k_1qL^2=\left(11 \times 10^{-5}\right) (4.2) L^2 \text{ gO}_2/\text{hr}$, and it will consume oxygen at a rate of
    $C=k_2V(L)=k_2pL^3 = \left(3.8 \times 10^{-5}\right)(0.3)L^3 \text{ gO}_2/\text{hr}$
    These rates balance when $A=C$, that is:
    $\begin{align*} A&=C\\ \left(11 \times 10^{-5}\right) (4.2) L^2&= \left(3.8 \times 10^{-5}\right)(0.3)L^3\\ 11 \times 4.2&=3.8 \times 0.3L\\ L&=41+\frac{28}{57}\\ &\approx 41.5 \text{ cm} \end{align*}$
  3. A frog of length $L$ will abosorb oxygen at a rate of
    $A=k_1S(L) = k_1qL^2\text{ gO}_2/\text{hr}$, and it will consume oxygen at a rate of
    $C=k_2V(L)=k_2pL^3 \text{ gO}_2/\text{hr}$
    These rates balance when $A=C$, that is:
    $\begin{align*} A&=C\\ k_1qL^2&=k_2pL^3\\ L&=\frac{k_1q}{k_2p} \text{ cm} \end{align*}$
  4. Note that the absorbtion of rate of oxygen, $A(L)=(k_1q)L^2$, is a quadratic power function of $L$, while the consumption rate of oxygen, $C(L)=(k_2p)L^3$, is a cubic power function of $L$. So, when a frog has length larger than $L_{\text{bal}}$, we expect the rate of oxygen consumption to be faster than the rate of absorption: that is, the frog cannot stay underwater for an extended period of time without suffocating. On the other hand, when a frog has length less than $L_{\text{bal}}$, the model predicts that its absorption will outpace its consumption, and it will take in more oxygen than it needs. FrogGraph.png
  5. Allocation of points

    1. (1 point) Prediction should be clearly stated.
      • (1 point) Equation for absorption, with all constants turned into numbers.
      • (1 point) Equation for consumption, with all constants turned into numbers.
      • (1 point) Setting equal, solving
    2. (1 point) Correct answer, with correct constants.
    3. Explanation:
      • (1 point) If $L>L_{\text{bal}}$, consumption is larger, the amount of oxygen in the frog decreases, and the frog suffocates if it doesn't surface.
      • (1 point) If $L<L_{\text{bal}}$, absorption is larger, so the amount of oxygen in the frog increases (beyond the frog's needs).
      Graph:
      • (1 point) $L_{\text{bal}}$ is the $L$-value where the two curves meet
      • (1 point) The two curves have power-function shape, with $C(L)$ flatter to the left of $L_{\text{bal}}$, and steeper to the right.

    Communication: 0, 1 or 2 pts. 0 pts if the work is hard to follow because of a lack or excess of verbal explanation. 1 pt if it is ok to follow but has some missing or extraneous verbal explanation. 2 pts if it is both complete and concise in the verbal explanations. See examples online for guidance.

    Presentation: 0 or 1 pt. 0 pts might be given if the presentation (rather than the communication) is an obstacle to following your work. This can include poor image quality, disorganized use of space etc.

    Total: 12 pts.