TE
Thomas E.

### Problem 38

The rate of heat loss (in watts) in harbor seal pups has been 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 $\left(\text { in }^{\circ} \mathrm{C}\right) .$ Find the heat loss for the following data. Source: Functional Ecology.
a. Body mass $=21 \mathrm{kg} ;$ body core temperature $=36^{\circ} \mathrm{C} ;$ ambient water temperature $=4^{\circ} \mathrm{C}$
b. Body mass $=29 \mathrm{kg} ;$ body core temperature $=38^{\circ} \mathrm{C} ;$ ambient water temperature $=16^{\circ} \mathrm{C}$

TE
Thomas E.

### Problem 39

The surface area of a human (in square meters) has been approximated by
$$A=0.024265 h^{0.3964} m^{0.5378}$$
where $h$ is the height $(\mathrm{in} \mathrm{cm})$ and $m$ is the mass (in $\mathrm{kg} ) .$ Find $A$ for the following data. Source: The Journal of Pediatrics.
a. Height, $178 \mathrm{cm} ;$ mass, 72 $\mathrm{kg}$
b. Height, $140 \mathrm{cm} ;$ mass, 65 $\mathrm{kg}$
c. Height, $160 \mathrm{cm} ;$ mass, 70 $\mathrm{kg}$

TE
Thomas E.

### Problem 40

An article entitled “How Dinosaurs Ran” explains that the locomotion of different sized animals can be compared when they have the same Froude number, defined as
$$F=\frac{v^{2}}{g l}$$
where $v$ is the velocity, $g$ is the acceleration of gravity $\left(9.81 \mathrm{m} \text { per } \sec ^{2}\right),$ and $l$ is the leg length (in meters). Source: Scientific American.

a. One result described in the article is that different animals change from a trot to a gallop at the same Froude number, roughly 2.56. Find the velocity at which this change occurs for a ferret, with a leg length of 0.09 m, and a rhinoceros, with a leg length of 1.2 m.
b. Ancient footprints in Texas of a sauropod, a large herbivo- rous dinosaur, are roughly 1 m in diameter, corresponding to a leg length of roughly 4 m. By comparing the stride divided by the leg length with that of various modern creatures, it can be determined that the Froude number for these dinosaurs is roughly 0.025. How fast were the sauropods traveling?

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### Problem 41

According to research at the Great Swamp in New York, the percentage of fish that are intolerant to pollution can be estimated by the function
$$P(W, R, A)=48-2.43 W-1.81 R-1.22 A$$
where $W$ is the percentage of wetland, $R$ is the percentage of residential area, and $A$ is the percentage of agricultural area surrounding the swamp. Source: Northeastern Naturalist.

a. Use this function to estimate the percentage of fish that will be intolerant to pollution if 5 percent of the land is classified as wetland, 15 percent is classified as residential, and 0 percent is classified as agricultural. (Note: The land can also be classified as forest land.)
b. What is the maximum percentage of fish that will be intolerant to pollution?
c. Develop two scenarios that will drive the percentage of fish that are intolerant to pollution to zero.
d. Which variable has the greatest influence on P?

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### Problem 42

In tropical regions, dengue fever is a significant health problem that affects nearly 100 million people each year. Using data collected from the 2002 dengue epidemic in Colima, Mexico, researchers have estimated that the incidence $I$ (number of new cases in a given year) of dengue can be predicted by the following function.
\begin{aligned} I(p, a, m, n, e)=&(25.54+0.04 p-7.92 a+2.62 m\\ &+4.46 n+0.15 e )^{2} \end{aligned}
where $p$ is the precipitation $(\mathrm{mm}), a$ is the mean temperature $\left(^{\circ} \mathrm{C}\right), m$ is the maximum temperature $\left(^{\circ} \mathrm{C}\right), n$ is the minimum temperature $\left(^{\circ} \mathrm{C}\right),$ and $e$ is the evaporation $(\mathrm{mm}) .$ Source: Journal of Environmental Health.
a. Estimate the incidence of a dengue fever outbreak for a region with 80 $\mathrm{mm}$ of rainfall, average temperature of $23^{\circ} \mathrm{C}$ , maximum temperature of $34^{\circ} \mathrm{C}$ , minimum temperature of $16^{\circ} \mathrm{C},$ and evaporation of 50 $\mathrm{mm} .$
b. Which variable has a negative influence on the incidence of dengue? Describe this influence and what can be inferred mathematically about the biology of the fever.

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### Problem 43

Using data collected by the U.S. Forest Service, the annual number of deer-vehicle accidents for any given county in Ohio can be estimated by the function
\begin{aligned} A(L, T, U, C)=& 53.02+0.383 L+0.0015 T+0.0028 U \\ &-0.0003 C \end{aligned}
where $A$ is the estimated number of accidents, $L$ is the road length (in kilometers), $T$ is the total county land area (in hundred-acres (Ha)), $U$ is the urban land area (in hundred- acres), and $C$ is the number of hundred-acres of crop land. Source: Ohio Journal of Science.
a. Use this formula to estimate the number of deer-vehicle accidents for Mahoning County, where $L=266 \mathrm{km}, T=$ $107,484 \mathrm{Ha}, U=31,697 \mathrm{Ha},$ and $C=24,870$ Ha. The actual value was $396 .$
b. Given the magnitude and nature of the input numbers, which of the variables have the greatest potential to influence the number of deer-vehicle accidents? Explain your answer.

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### Problem 44

Using data collected by the U.S. Forest Service, the annual number of deer that are harvested for any given county in Ohio can be estimated by the function
$$N(R, C)=329.32+0.0377 R-0.0171 C$$
where $N$ is the estimated number of harvested deer, $R$ is the rural land area (in hundred-acres), and $C$ is the number of hundred-acres of crop land. Source: Ohio Journal of Science.
a. Use this formula to estimate the number of harvested deer for Tuscarawas County, where $R=141,319$ Ha and $\mathrm{C}=$ $37,960$ Ha. The actual value was 4925 deer harvested.
b. Sketch the graph of this function in the first octant.

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### Problem 45

Pregnant sows tethered in stalls often show high levels of repetitive behavior, such as bar biting and chain chewing, indicating chronic stress. Researchers from Great Britain have developed a function that estimates the relationship between repetitive behavior, the behavior of sows in adjacent stalls, and food allowances such that
$$\ln (T)=5.49-3.00 \ln (F)+0.18 \ln (C)$$
where $T$ is the percent of time spent in repetitive behavior, $F$ is the amount of food given to the sow (in kilograms per day), and $C$ is the percent of time that neighboring sows spent bar biting and chain chewing. Source: Applied Animal Behaviour Science.
a. Solve the above expression for $T$
b. Find and interpret $T$ when $F=2$ and $C=40$ .

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### Problem 46

Extra postage is charged for parcels sent by U.S. mail that are more than 84 in. in length and girth combined. (Girth is the distance around the parcel perpendicular to its length. See the figure.) Express the combined length and girth as a function of $L, W,$ and $H$

TE
Thomas E.

### Problem 47

Refer to the figure for Exercise $46 .$ Assume $L, W,$ and $H$ are in feet. Write a function in terms of $L, W,$ and $H$ that gives the total area of the material required to build the box.

TE
Thomas E.
$$L=f(H, D)=\sqrt{H^{2}+D^{2}}$$
where $D$ is the (outside) diameter of the pipe and $H$ is the "rise "of the roof per $D$ units of "run'; that is, the slope of the roof is $H / D$ . (See the figure below.) The width of the ellipse (minor axis) equals $D .$ Find the length and width of the ellipse required to produce a hole for a vent pipe with a diameter of 3.75 in. in roofs with the following slopes.
a. 3$/ 4$
b. 2$/ 5$