1. **State the problem:** We want to find how fast the brain weight $B$ of a fish species is growing when the average length $L$ is 18 cm, given that $B$ depends on body weight $W$, and $W$ depends on length $L$. The length changes over time at a constant rate. 2. **Given formulas:** - Brain weight as a function of body weight: $$B = 0.007 W^{\frac{2}{3}}$$ - Body weight as a function of length: $$W = 0.12 L^{2.53}$$ 3. **Find the rate of change of brain weight with respect to time $t$: $$\frac{dB}{dt}$$ when $L=18$ cm.** 4. **Use the chain rule:** $$\frac{dB}{dt} = \frac{dB}{dW} \cdot \frac{dW}{dL} \cdot \frac{dL}{dt}$$ 5. **Calculate each derivative:** - $$\frac{dB}{dW} = 0.007 \cdot \frac{2}{3} W^{-\frac{1}{3}} = \frac{0.007 \times 2}{3} W^{-\frac{1}{3}} = 0.0046667 W^{-\frac{1}{3}}$$ - $$\frac{dW}{dL} = 0.12 \times 2.53 L^{2.53 - 1} = 0.3036 L^{1.53}$$ 6. **Find $W$ at $L=18$ cm:** $$W = 0.12 \times 18^{2.53}$$ Calculate $18^{2.53}$: $$\ln(18) \approx 2.8904$$ $$18^{2.53} = e^{2.53 \times 2.8904} = e^{7.311} \approx 1499.5$$ So, $$W = 0.12 \times 1499.5 = 179.94 \text{ grams}$$ 7. **Calculate $\frac{dB}{dW}$ at $W=179.94$:** $$W^{-\frac{1}{3}} = (179.94)^{-\frac{1}{3}} = \frac{1}{(179.94)^{1/3}}$$ Calculate cube root: $$\sqrt[3]{179.94} \approx 5.62$$ So, $$W^{-\frac{1}{3}} \approx \frac{1}{5.62} = 0.178$$ Then, $$\frac{dB}{dW} = 0.0046667 \times 0.178 = 0.00083$$ 8. **Calculate $\frac{dW}{dL}$ at $L=18$:** $$\frac{dW}{dL} = 0.3036 \times 18^{1.53}$$ Calculate $18^{1.53}$: $$18^{1.53} = e^{1.53 \times 2.8904} = e^{4.423} \approx 83.7$$ So, $$\frac{dW}{dL} = 0.3036 \times 83.7 = 25.4$$ 9. **Find $\frac{dL}{dt}$:** Length changes from 15 cm to 20 cm over 10 million years. $$\frac{dL}{dt} = \frac{20 - 15}{10,000,000} = \frac{5}{10,000,000} = 5 \times 10^{-7} \text{ cm/year}$$ 10. **Calculate $\frac{dB}{dt}$:** $$\frac{dB}{dt} = 0.00083 \times 25.4 \times 5 \times 10^{-7} = 0.00083 \times 25.4 \times 5 \times 10^{-7}$$ Calculate stepwise: $$0.00083 \times 25.4 = 0.0211$$ $$0.0211 \times 5 \times 10^{-7} = 1.055 \times 10^{-8} \text{ grams/year}$$ 11. **Convert grams/year to nanograms/year:** $$1 \text{ gram} = 10^{9} \text{ nanograms}$$ So, $$\frac{dB}{dt} = 1.055 \times 10^{-8} \times 10^{9} = 10.55 \text{ nanograms/year}$$ **Final answer:** The brain weight was growing at approximately **10.55 nanograms per year** when the average length was 18 cm.