Chapter 3.3, Problem 43PS

Solution

Simplify $r(w)$ and calculate the breathing rate for body weights 6.5 grams, 300 grams and 70,000 grams.

  $r(w)=220w^{-0.266}$

For w = 6.5 gram, 134 breaths per minute

For w = 300 gram, 48.4 breaths per minute

For w = 70000 gram, 11.3 breaths per minute

Given:

For a mammal that weighs w grams, the volume b (in milliliters) of air breathed in and the volume d (in milliliter) of “dead space” (the portion of the lungs not filled with air) can be modeled by:

  $b(w)=0.007w$

  $d(w)=0.002w$

The breathing rate r (in breaths per minute) of a mammal that weighs w in grams can be modeled by:

  $r(w)=\frac{1.1w^{0.734}}{b(w)-d(w)}$

Concept Used:

• We have two functions, f(x) and g(x). We can define the sum of these two functions by,  $(f+g)\left(x\right)$= $f\left(x\right)$+ $g\left(x\right)$, where x is in the domain of both f and g.
• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

  $b(w)=0.007w$

  $d(w)=0.002w$

  $\begin{array}{l}r(w)=\frac{1.1w^{0.734}}{b(w)-d(w)}\\ r(w)=\frac{1.1w^{0.734}}{0.007w-0.002w}\\ r(w)=\frac{1.1w^{0.734}}{0.005w}\\ r(w)=220w^{0.734}(w^{-1})\\ r(w)=220w^{0.734-1}\\ r(w)=220w^{-0.266}\end{array}$

For w = 6.5 gram

  $\begin{array}{c}r(w)=220w^{-0.266}\\ $ =220{(6.5)}^{-0.266}$=220(0.6078)\\ =133.716\\ \sim 134\end{array}$

For w = 6.5 gram, 134 breaths per minute

For w = 300 grams-

  $\begin{array}{c}r(w)=220w^{-0.266}\\ $ =220{(300)}^{-0.266}$=220(0.22)\\ =48.4\end{array}$

For w = 300 gram, 48.4 breaths per minute

For w = 70,000 grams-

  $\begin{array}{c}r(w)=220w^{-0.266}\\ $ =220{(70000)}^{-0.266}$=220(0.0514)\\ =11.30\end{array}$

For w = 70000 gram, 11.3 breaths per minute