Question

for a particular salmon population, the relationship between the number s of spawners and the number r of offspring that survive to maturity is given by the formula r = \frac{4100s}{s + 800}. (a) under what conditions is r > s? 0 < s < 4900; 0 < s < 3300; -800 < s < 0; -3300 < s < 0; 0 < s < 4100. (b) find the number of spawners that would yield 90% of the greatest possible number of offspring that survive to maturity. spawners. (c) work part (b) with 80% replacing 90%. spawners. (d) compare the results for s and r (in terms of percentage increases) from parts (b) and (c). a 225% increase in s produces a 112.5% increase in r; a 125% increase in s produces a 12.5% increase in r; a 225% increase in s produces a 12.5% increase in r; a 125% increase in s produces a 112.5% increase in r.

Explanation:

Step1: Solve for when $R > S$

We have $R=\frac{4100S}{S + 800}$, and we want $R>S$. So $\frac{4100S}{S + 800}>S$. Since $S$ represents the number of spawners, $S>0$ (non - negative population). First, multiply both sides by $S + 800$ (assuming $S>- 800$, which is true for non - negative $S$) to get $4100S>S(S + 800)$. Expand the right side: $4100S>S^{2}+800S$. Rearrange to get a quadratic inequality: $S^{2}-3300S<0$. Factor out $S$: $S(S - 3300)<0$. The solutions of the corresponding quadratic equation $S(S - 3300)=0$ are $S = 0$ and $S=3300$. The inequality is satisfied when $0

Step2: Find the maximum of $R$

We can rewrite $R=\frac{4100S}{S + 800}=\frac{4100(S + 800-800)}{S + 800}=4100-\frac{4100\times800}{S + 800}$. As $S$ gets very large, $R$ approaches $4100$.

Step3: Find the number of spawners for 90% of the maximum

We want $R = 0.9\times4100 = 3690$. Set $\frac{4100S}{S + 800}=3690$. Cross - multiply: $4100S=3690(S + 800)$. Expand: $4100S=3690S+3690\times800$. Subtract $3690S$ from both sides: $4100S-3690S=3690\times800$. So $410S=3690\times800$. Solve for $S$: $S=\frac{3690\times800}{410}=7200$.

Step4: Find the number of spawners for 80% of the maximum

We want $R = 0.8\times4100 = 3280$. Set $\frac{4100S}{S + 800}=3280$. Cross - multiply: $4100S=3280(S + 800)$. Expand: $4100S=3280S+3280\times800$. Subtract $3280S$ from both sides: $4100S - 3280S=3280\times800$. So $820S=3280\times800$. Solve for $S$: $S=\frac{3280\times800}{820}=3200$.

Step5: Calculate percentage increases

The increase in $S$ from part (c) to part (b) is $\frac{7200 - 3200}{3200}\times100\%=\frac{4000}{3200}\times100\% = 125\%$. The increase in $R$ from part (c) to part (b) is $\frac{3690 - 3280}{3280}\times100\%=\frac{410}{3280}\times100\%=12.5\%$.

Answer:

(a) 0 < S < 3300
(b) 7200
(c) 3200
(d) A 125% increase in S produces a 12.5% increase in R.