Coroners estimate the time of death using the rule of thumb that a body cools about $2^{\circ} \mathrm{F}$ during the first hour after death and about $1^{\circ} \mathrm{F}$ for each additional hour. Assuming an air temperature of $68^{\circ} \mathrm{F}$ and a living body temperature of $98.6^{\circ} \mathrm{F}$, the temperature $T(t)$ in ${ }^{\circ} \mathrm{F}$ of a body at a time $t$ hours since death is given by

$$T(t)=68+30.6 e^{-k t} .$$

$\bull$ Using the value of $k$ found in part (a), show that, 24 hours after death, the coroner's rule of thumb gives approximately the same temperature as the formula.

Coroners estimate the time of death using the rule of thumb that a body cools about $2^{\circ} \mathrm{F}$ during the first hour after death and about $1^{\circ} \mathrm{F}$ for each additional hour. Assuming an air temperature of $68^{\circ} \mathrm{F}$ and a living body temperature of $98.6^{\circ} \mathrm{F}$, the temperature $T(t)$ in ${ }^{\circ} \mathrm{F}$ of a body at a time $t$ hours since death is given by

$$T(t)=68+30.6 e^{-k t} .$$

$\bull$ For what value of $k$ will the body cool by $2^{\circ} \mathrm{F}$ in the first hour?

Coroners estimate time of death using the rule of thumb that a body cools about $2 ^ { \circ } \mathrm { F }$ during the first hour after death and about $1 ^ { \circ } \mathrm { F }$ for each additional hour. Assuming an air temperature of $68 ^ { \circ } \mathrm { F }$ and a living body temperature of $98.6 ^ { \circ } \mathrm { F },$ the temperature T(t) in $^ { \circ } \mathrm { F }$ of a body at a time t hours since death is given by $T ( t ) = 68 + 30.6 e ^ { - k t }.$ (a) For what value of k will the body cool by $2 ^ { \circ } \mathrm { F }$ in the first hour? (b) Using the value of k found in part (a), after how many hours will the temperature of the body be decreasing at a rate of $1 ^ { \circ } \mathrm { F }$ per hour? (c) Using the value of k found in part (a), show that, 24 hours after death, the coroner's rule of thumb gives approximately the same temperature as the formula.

Coroners estimate time of death using the rule of thumb that a body cools about $2^{\circ} \mathrm{F}$ during the first hour after death and about $1^{\circ} \mathrm{F}$ for each additional hour. Assuming an air temperature of $68^{\circ} \mathrm{F}$ and a living body temperature of $98.6^{\circ} \mathrm{F}$, the temperature T(t) in $^{\circ} \mathrm{F}$ of a body at a time t hours since death is given by $T(t)=68+30.6 e^{-k t}$. For what value of k will the body cool by $2^{\circ} \mathrm{F}$ in the first hour?

Evaluate the limit where $f ( t ) = \left( \frac { 3 ^ { t } + 5 ^ { t } } { 2 } \right) ^ { 1 / t }$ for $t \neq 0. \quad \lim _ { t \rightarrow 0 } f ( t )$

A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?

Gasoline is pouring into a vertical cylindrical tank of radius 3 feet. When the depth of the gasoline is 4 feet, the depth is increasing at 0.2 ft/sec. How fast is the volume of gasoline changing at that instant?

A thin uniform rod of length $l$ cm and a small particle lie on a line separated by a distance of $a$ cm. If $K$ is a positive constant and $F$ is measured in newtons, the gravitational force between them is

$$F=\frac{K}{a(a+l)}.$$

(a) If $a$ is increasing at the rate $2$ cm/min when $a = 15$ and $l = 5$, how fast is $F$ decreasing?

A thin uniform rod of length $l$ cm and a small particle lie on a line separated by a distance of $a$ cm. If $K$ is a positive constant and $F$ is measured in newtons, the gravitational force between them is

$$F=\frac{K}{a(a+l)}.$$

(b) If $l$ is decreasing at the rate $2$ cm/min when $a = 15$ and $l = 5$, how fast is $F$ increasing?

Car A is driving east toward an intersection. Car B has already gone through the same intersection and is heading north. At what rate is the distance between the cars changing at the instant when car A is 40 miles from the intersection and traveling at 50 mph and car B is 30 miles from the intersection and traveling at 60 mph? Are the cars getting closer together or farther apart at this time?

A thin uniform rod of length l cm and a small particle lie on a line separated by a distance of a cm. If K is a positive constant and F is measured in newtons, the gravitational force between them is $F = \frac { K } { a ( a + l ) }.$ (a) If a is increasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F decreasing? (b) If l is decreasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F increasing?

A voltage V across a resistance R generates a current $I = \frac { V } { R }$ in amps. A constant voltage of 9 volts is put across a resistance that is increasing at a rate of 0.2 ohms per second when the resistance is 5 ohms. At what rate is the current changing?

The DuBois formula relates a person's surface area s, in $\text {m} ^ 2,$ to weight w, in kg, and height h, in cm, by $s = 0.01 w ^ { 0.25 } h ^ { 0.75 }.$ (a) What is the surface area of a person who weighs 65 kg and is 160 cm tall? (b) What is the weight of a person whose height is 180 cm and who has a surface area of $1.5 \text {m}^2 ?$ (c) For people of fixed weight 70 kg, solve for h as a function of s. Simplify your answer.

The Dubois formula relates a person's surface area, s, in meters $^2,$ to weight, w, in kg, and height, h, in cm, by $s = 0.01 w ^ { 0.25 } h ^ { 0.75 }.$ (a) What is the surface area of a person who weighs 60 kg and is 150 cm tall? (b) The person in part (a) stays constant height but increases in weight by 0.5 kg/year. At what rate is his surface area increasing when his weight is 62 kg?