# OSH/1/Solution

< OSH‎ | 1

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.
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.