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(t), in degrees Fahrenheit, of the aluminum t minutes after it is removed from the oven.

t = Minutes A = Temperature

0 306

30 150

60 104

90 83

120 75

150 73

180 72

210 72

(a) Explain the meaning of A(75).

The expression A(75) is the amount of time it takes for the aluminum to reach 75 degrees Fahrenheit.The expression A(75) is the temperature of the piece of aluminum when the temperature of the oven reaches 75 degrees Fahrenheit.    The expression A(75) is the amount of time it takes for the aluminum to reach 75 degrees Fahrenheit after removed from the oven.The expression A(75) is the temperature in degrees Fahrenheit of the piece of aluminum 75 minutes after it is removed from the oven.

Estimate its value.
°F

(b) Find the average decrease per minute of temperature during the first half-hour of cooling. (Round your answer to two decimal places.)
° per minute

(c) Find the average decrease per minute of temperature during the first half of the second hour of cooling. (Round your answer to two decimal places.)
° per minute
(d) Explain how parts (b) and (c) support Newton's law of cooling. (Select all that apply.)

We can see from parts (b) and (c) that the aluminum cools at a constant rate.We can see from parts (b) and (c) that the aluminum cools at a faster rate early and then cools more slowly later.We can see from parts (b) and (c) that the aluminum cools at a slower rate early and then cools more quickly later.This shows that the object cools more rapidly when the difference between the object's temperature and the surrounding temperature is large.This shows that the object cools more rapidly when the difference between the object's temperature and the surrounding temperature is small.

(e) Use functional notation to express the temperature of the aluminum after 1 hour and 11 minutes. Estimate the temperature at that time. (Note: Your work in part (c) should be helpful. Round your answer to two decimal places.)
A(  ) =  °F

(f) What is the temperature of the oven? Express your answer using functional notation, and give its value.
A(  ) =  °F
(g) Explain why you would expect the function A to have a limiting value.

This answer has not been graded yet.

(h) What is the temperature in the room?
°F

Explain your reasoning.

Name: Newton's law of cooling says that a hot object cools rapidly when the
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Description: 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

Question

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(t), in degrees Fahrenheit, of the aluminum t minutes after it is removed from the oven.

t = Minutes	A = Temperature
0	306
30	150
60	104
90	83
120	75
150	73
180	72
210	72

(a) Explain the meaning of A(75).

The expression A(75) is the amount of time it takes for the aluminum to reach 75 degrees Fahrenheit.The expression A(75) is the temperature of the piece of aluminum when the temperature of the oven reaches 75 degrees Fahrenheit. The expression A(75) is the amount of time it takes for the aluminum to reach 75 degrees Fahrenheit after removed from the oven.The expression A(75) is the temperature in degrees Fahrenheit of the piece of aluminum 75 minutes after it is removed from the oven.

Estimate its value.
°F

(b) Find the average decrease per minute of temperature during the first half-hour of cooling. (Round your answer to two decimal places.)
° per minute

(c) Find the average decrease per minute of temperature during the first half of the second hour of cooling. (Round your answer to two decimal places.)
° per minute
(d) Explain how parts (b) and (c) support Newton's law of cooling. (Select all that apply.)

We can see from parts (b) and (c) that the aluminum cools at a constant rate.We can see from parts (b) and (c) that the aluminum cools at a faster rate early and then cools more slowly later.We can see from parts (b) and (c) that the aluminum cools at a slower rate early and then cools more quickly later.This shows that the object cools more rapidly when the difference between the object's temperature and the surrounding temperature is large.This shows that the object cools more rapidly when the difference between the object's temperature and the surrounding temperature is small.

(e) Use functional notation to express the temperature of the aluminum after 1 hour and 11 minutes. Estimate the temperature at that time. (Note: Your work in part (c) should be helpful. Round your answer to two decimal places.)
A( ) = °F

(f) What is the temperature of the oven? Express your answer using functional notation, and give its value.
A( ) = °F
(g) Explain why you would expect the function A to have a limiting value.

This answer has not been graded yet.

(h) What is the temperature in the room?
°F

Explain your reasoning.

Accepted Answer

Answered: Image /qna-images/answer/09832384-b2aa-4c21-85c7-a407d2bbcfdf.jpg