The energy cost of a speed burst as a function of the body weight of a dolphin is given by E = 43. 5w-0. 61, where w is the weight of the dolphin (in kg) and E is the energy expenditure (in kcal/kg/km). Suppose that the weight of a 400-kg dolphin is increasing at a rate of 8 kg/day. Find the rate at which the energy expenditure is changing with respect to time. A) -0. 0017 kcal/kg/km/day B) -20. 5166 kcal/kg/km/day C) -0. 0137 kcal/kg/km/day D) -5. 491 kcal/kg/km/day

The rate at which the energy expenditure is changing with respect to time is -0.0137 kcal/kg/km/day.

To find the rate at which the energy expenditure is changing with respect to time, we need to use the chain rule of differentiation.

Given the equation E = 43.5w^(-0.61), where E represents energy expenditure and w represents the weight of the dolphin in kg, we want to find dE/dt, the rate of change of energy expenditure with respect to time.

First, we express w as a function of time t. We are given that the weight of the dolphin is increasing at a rate of 8 kg/day, so we can write w = 400 + 8t.

Now, we differentiate E with respect to t:

dE/dt = dE/dw * dw/dt

To find dE/dw, we differentiate E with respect to w:

dE/dw = -0.61 * 43.5 * w^(-0.61 - 1) = -26.5735 * w^(-1.61)

Substituting w = 400 + 8t:

dE/dw = -26.5735 * (400 + 8t)^(-1.61)

Next, we find dw/dt:

dw/dt = 8

Finally, we can calculate dE/dt:

dE/dt = -26.5735 * (400 + 8t)^(-1.61) * 8

Evaluating this expression at t = 0 (initial time), we get:

dE/dt = -26.5735 * (400 + 8 * 0)^(-1.61) * 8 = -26.5735 * 400^(-1.61) * 8

Simplifying the expression yields:

dE/dt ≈ -0.0137 kcal/kg/km/day

Therefore, the rate at which the energy expenditure is changing with respect to time is approximately -0.0137 kcal/kg/km/day.

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