dm/dt= (.54*85.54/w.46)*dw/dt
solve for dm/dt when w=.25 and dw/dt=.01kg/day
solve for dm/dt when w=.25 and dw/dt=.01kg/day
Given:
Metabolic rate function: m = 85.65w^0.54
Weight: w = 0.25 kg
Weight change rate: dW/dt = 0.01 kg/day
Let's differentiate the metabolic rate function with respect to time (t):
dm/dt = (d/dt) (85.65w^0.54)
To find this derivative, we need to use the chain rule. The chain rule states that if we have a composite function, we need to differentiate the outer function and multiply by the derivative of the inner function. In this case, the outer function is (85.65w^0.54) and the inner function is w.
First, let's find the derivative of the outer function:
d/dt (85.65w^0.54)
= 85.65 * d/dt (w^0.54)
Next, let's find the derivative of the inner function:
d/dt (w^0.54)
= 0.54 * (d/dt)(w)
Remember that the weight is changing with time, so we have to consider dW/dt:
= 0.54 * dW/dt
Finally, we can substitute the values into our equation:
dm/dt = 85.65 * 0.54 * dW/dt
Given that dW/dt = 0.01 kg/day:
dm/dt = 85.65 * 0.54 * 0.01
= 0.462381 kcal/day
Therefore, the rate of change in the metabolic rate with respect to time for a 0.25 kg rodent gaining weight at a rate of 0.01 kg/day is 0.462381 kcal/day.