Chapter 1.2, Problem 14E

Reminder Round all answers to two decimal places unless otherwise indicated.

Newton's Law of Cooling says that a hot object cools rapidly when the difference between its temperature and that of the surrounding air is large, but the object cools more slowly when it nears room temperature. Suppose a piece of aluminum is removed from an oven and left to cool. The following table gives the temperature $A=A\left(t\right)$, in degrees Fahrenheit. of the aluminum $t$ minutes after it is removed from the own.

a. Explain the meaning of $A\left(75\right)$ and estimate its value.

b. Find the average decrease of temperature pet minute during the first half-hour of cooling.

$t=\text{Minutes}$	$A=\text{Temperature}$
$0$	$302$
$30$	$152$
$60$	$100$
$90$	$81$
$120$	$75$
$150$	$73$
$180$	$72$
$210$	$72$

c. Find the average decrease per minute of temperature during the first half of the second hour of cooling.

d. Explain how parts b and c support Newton's law of cooling.

e. Use functional notation to express the temperature of the aluminum after I hour and 13 minutes. Estimate the temperature at that time. (Note: Your work in part c should be helpful.)

f. What is the temperature of the oven? Exams your answer using functional notation, and give its value.

g. Explain why you would expect the function A to have a limiting value.

h. What is roan temperature? Explain your reasoning.