Heat Loss In Exercise 56 of Section 2 of this chapter, we...

Heat Loss In Exercise 56 of Section 2 of this chapter, we found that the rate of heat loss (in watts) in harbor seal pups could be approximated by $$H(m, T, A)=\frac{15.2 m^{0.67}(T-A)}{10.23 \ln m-10.74}$$ where $m$ is the body mass of the pup (in $\mathrm{kg} ),$ and $T$ and $A$ are the body core temperature and ambient water temperature, respectively $\left(\mathrm{in}^{\circ} \mathrm{C}\right) .$ Suppose $m$ is $25 \mathrm{kg}, T$ is $36.0^{\circ},$ and $A$ is $12.0^{\circ} \mathrm{C}$ . Approximate the change in $H$ if $m$ changes to $26 \mathrm{kg}, T$ to $36.5^{\circ},$and $A$ to $10.0^{\circ} \mathrm{C} .$

Heat Loss In Exercise 56 of Section 2 of this chapter, we found that the rate of heat loss (in watts) in harbor seal pups could be approximated by
$$H(m, T, A)=\frac{15.2 m^{0.67}(T-A)}{10.23 \ln m-10.74}$$
where $m$ is the body mass of the pup (in $\mathrm{kg} ),$ and $T$ and $A$ are the body core temperature and ambient water temperature, respectively $\left(\mathrm{in}^{\circ} \mathrm{C}\right) .$ Suppose $m$ is $25 \mathrm{kg}, T$ is $36.0^{\circ},$ and $A$ is $12.0^{\circ} \mathrm{C}$ . Approximate the change in $H$ if $m$ changes to $26 \mathrm{kg}, T$ to $36.5^{\circ},$and $A$ to $10.0^{\circ} \mathrm{C} .$

Key Concepts

Partial Derivatives

Partial derivatives measure the rate of change of a function with respect to one variable at a time, while all other variables are held constant. They are fundamental in understanding how each independent variable contributes to changes in a multivariable function, which is essential in problems that involve calculating changes due to variations in multiple inputs.

Total Differential

The total differential of a multivariable function provides a linear approximation of the change in the function for small changes in its variables. By summing the product of each variable’s partial derivative with its small change, the total differential offers an efficient way to estimate the overall change in the function without recalculating the function exactly.

Linear Approximation

Linear approximation involves using the tangent plane at a specific point to approximate the value of a function near that point. This technique relies on the function's first derivatives and is widely used in problems where small perturbations in the input variables lead to an approximate change in the function, simplifying complex calculations.

Transcript

00:01 Okay, so we have this function.

00:04 H is a function of n, t, and a.

00:11 It's 15 .2, m to the 0 .6, 7 times t minus a, all over 10 .23 natural log of m minus 10 .74.

00:40 Okay, so h is the rate of heat loss in harbor seal pups in watts, and m is the mass of the body of the pup.

00:55 T is the core body temperature.

00:58 A is the ambient water temperature.

01:00 Okay, so this is the rate of heat loss when the pup is in water, ambient temperature of a okay so we're given that n is 25 t is 36 and that's degrees c and a it's 12 degrees celsius again h is in watts let's see what is mass in masses in kilograms, okay? masses in kilograms.

01:49 All right.

01:50 And so what we want to know is actually the change in this rate of heat loss.

01:59 If the mass changes to 26, so dm is 1, the core temperature dt changes to 36 .5, so dt is 0 .5.

02:15 And the ambient temperature changes to 10.

02:18 So it's negative 2 for da.

02:21 Okay, and so we need all of the partial derivatives, and we'll multiply those by the differentials.

02:28 So let's look at, say, hm.

02:32 I think this is going to be the most complicated one.

02:35 Okay, so we have a quotient rule.

02:40 Okay, let's see.

02:41 How do we want to do this? so we have the bottom times 0 .6 ,7.

02:59 Times 15 .2 m to the negative 0 .33, t minus a.