The average daily metabolic rate for rodents can be expressed by the function m=85.65w^0.54, where w is the weight in kg and m is the metabolic rate in kcal/day. Determine the rate of change in the metabolic rate with respect to time for a 0.25kg rodent that is gaining weight at a rate T 0.01 kg/day
dm/dt= (.54*85.54/w.46)*dw/dt
solve for dm/dt when w=.25 and dw/dt=.01kg/day
To determine the rate of change in the metabolic rate with respect to time, we need to find the derivative of the metabolic rate function with respect to time.
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.