For many species of fish, the allometric relationship between the weight W and the length L is approximately where k is a constant. For find the percent rate of change of the weight as a corresponding percent rate of change of the length.

Question

For many species of fish, the allometric relationship between the weight W and the length L is approximately  where k is a constant. For  find the percent rate of change of the weight as a corresponding percent rate of change of the length. ​

Quizplus · Accepted Answer

Taking the logarithm of both sides of the given equation, we get:

log(W) = log(k) + 3log(L)

Differentiating with respect to time t, we get:

d(log(W))/dt = 3d(log(L))/dt

Using the chain rule, we can express the left-hand side as:

d(log(W))/dt = (1/W)(dW/dt)

Similarly, the right-hand side can be expressed as:

d(log(L))/dt = (1/L)(dL/dt)

Substituting these expressions back into the previous equation, we get:

(1/W)(dW/dt) = 3(1/L)(dL/dt)

Simplifying and solving for dW/dt, we get:

dW/dt = (3W/L)(dL/dt)

This equation tells us that the percent rate of change of W is equal to 3 times the percent rate of change of L. Therefore, the correct answer is:

Choice A: 11ea984b_b1e6_9810_9029_c3fabc063054_TB4005_11

For Many Species of Fish, the Allometric Relationship Between the Weight