Question

Brain weight B as a function of body weight W in fish has been modeled by the power function B=.007W^(2/3), where B and W are measured in grams. A model for body weight as a function of body length L (measured in cm) is W=.12L^(2.53). If, over 10 million years, the average length of a certain species of fish evolved from 15cm to 20cm at a constant rate, how fast was the species' brain growing when the average length was 18cm? Round your answer to the nearest hundredth.

=......................nanograms/yr

Answer 1

Brain weight B as a function of body weight W in fish has been modeled by the power function B=.007W^(2/3), where B and W are measured in grams. A model for body weight as a function of body length L (measured in cm) is W=.12L^(2.53). If, over 10 million years, the average length of a certain species of fish evolved from 15cm to 20cm at a constant rate, how fast was the species' brain growing when the average length was 18cm? Round your answer to the nearest hundredth.

=10.4...nanograms/yr

Solution

The length is growing at a constant rate (this means the rate of change of L over time, dL/dt, is constant) from 15 to 20 cms over a period of 10^7 years or

$$\frac {d L}{d t} = \frac {5}{1 0 ^ {7}} = 5 * 1 0 ^ {- 7}$$ $$\frac {\mathrm {d} W}{\mathrm {d t}} = 0. 1 2 * 2. 5 3 \mathrm {L} ^ {1. 5 3} \frac {\mathrm {d L}}{\mathrm {d t}}$$

... taking the derivative using the power rule.

$$\frac {\mathrm {d} W}{\mathrm {d t}} = 0. 3 0 3 6 * (1 8) ^ {1. 5 3} * 5 * 1 0 ^ {- 7} = 1. 2 6 * 1 0 ^ {- 5}$$ $$W = 0. 1 2 * 1 8 ^ {2. 5 3} = 1. 8 * 1 0 ^ {2}$$

Now,

$$\frac {\mathrm {d} B}{\mathrm {d t}} = 0. 0 0 7 * \left(\frac {2}{3}\right) W ^ {- \frac {1}{3}} \frac {\mathrm {d} W}{\mathrm {d t}}$$

Substitute here what you've computed for W and dW/dt to find the desired dB/dt.

$$\begin{array}{l} \frac {d B}{d t} = 0. 0 0 7 * \left(\frac {2}{3}\right) * 0. 1 7 7 * 1. 2 6 * 1 0 ^ {- 5} = 0. 0 0 1 0 4 * 1 0 ^ {- 5} \\ = 1. 0 4 * 1 0 ^ {- 8} = 1 0. 4 \frac {\text {nanograms}}{\text {yr}} \\ \end{array}$$