Question

the value, \\(v(m)\\), of a comic book \\(m\\) months after publication has an average rate of change of -0.04 between \\(m = 36\\) and \\(m = 60\\). which statement must be true?

• the value of the comic book decreased by a total of \\$0.04 between \\(m = 36\\) and \\(m = 60\\).
• the value of the comic book increased by a total of \\$0.04 between \\(m = 36\\) and \\(m = 60\\).
• the value of the comic book decreased by an average of \\$0.04 each month between \\(m = 36\\) and \\(m = 60\\).
• the value of the comic book increased by an average of \\$0.04 each month between \\(m = 36\\) and \\(m = 60\\).

Explanation:

Analyze the given average rate of change

The average rate of change of the value \(V(m)\) between \(m = 36\) and \(m = 60\) is \(-0.04\). This represents the average change in value per month over this interval.

Interpret the sign and magnitude

A negative rate of change (\(-0.04\)) indicates a decrease in value. The magnitude of \(0.04\) means the value decreased by an average of \$0.04 each month over the specified period.

Evaluate the options

• Option 1 states a total decrease of \$0.04, which is incorrect because \(-0.04\) is the rate per month, not the total change.
• Option 2 states a total increase, which is incorrect.
• Option 3 states the value decreased by an average of \$0.04 each month, which matches the definition of the average rate of change.
• Option 4 states an average increase, which is incorrect due to the negative sign.

Answer:

• The value of the comic book decreased by a total of $0.04 between m = 36 and m = 60
• The value of the comic book increased by a total of $0.04 between m = 36 and m = 60
• The value of the comic book decreased by an average of $0.04 each month between m = 36 and m = 60 (Correct answer)
• The value of the comic book increased by an average of $0.04 each month between m = 36 and m = 60