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The value, [tex]\(V(m)\)[/tex], of a comic book [tex]\(m\)[/tex] months after publication has an average rate of change of -0.04 between [tex]\(m=36\)[/tex] and [tex]\(m=60\)[/tex]. Which statement must be true?

A. The value of the comic book decreased by a total of \[tex]$0.04 between \(m=36\) and \(m=60\).
B. The value of the comic book increased by a total of \$[/tex]0.04 between [tex]\(m=36\)[/tex] and [tex]\(m=60\)[/tex].
C. The value of the comic book decreased by an average of \[tex]$0.04 each month between \(m=36\) and \(m=60\).
D. The value of the comic book increased by an average of \$[/tex]0.04 each month between [tex]\(m=36\)[/tex] and [tex]\(m=60\)[/tex].

Sagot :

GinnyAnswer
To determine the correct statement, let's analyze the information given:

1. The value of the comic book, [tex]\( V(m) \)[/tex], has an average rate of change of -0.04 between [tex]\( m = 36 \)[/tex] and [tex]\( m = 60 \)[/tex].

Average rate of change is defined as the change in value divided by the change in time. Here, the average rate of change is [tex]\(-0.04\)[/tex] dollars per month.

2. The time interval is from [tex]\( m = 36 \)[/tex] to [tex]\( m = 60 \)[/tex]. So, the change in time ([tex]\( \Delta m \)[/tex]) is:
[tex]\[ \Delta m = m_{\text{end}} - m_{\text{start}} = 60 - 36 = 24 \text{ months} \][/tex]

3. The total change in value ([tex]\( \Delta V \)[/tex]) over this interval can be calculated using the average rate of change and the change in time:
[tex]\[ \Delta V = \text{rate of change} \times \Delta m = (-0.04) \times 24 = -0.96 \text{ dollars} \][/tex]

Hence, the total change in value of the comic book over this time period is [tex]\(-0.96\)[/tex] dollars, indicating a decrease in value.

Now, examining each statement:

1. "The value of the comic book decreased by a total of [tex]$0.04 between \( m = 36 \) and \( m = 60 \)." - This is incorrect because the total decrease in value is \(-0.96\) dollars, not \(-0.04\) dollars. 2. "The value of the comic book increased by a total of $[/tex]0.04 between [tex]\( m = 36 \)[/tex] and [tex]\( m = 60 \)[/tex]."
- This is incorrect because the comic book's value decreased, not increased.

3. "The value of the comic book decreased by an average of [tex]$0.04 each month between \( m = 36 \) and \( m = 60 \)." - This is correct because the average rate of change is \(-0.04\) dollars per month, indicating a decrease of \(-0.04\) dollars per month on average. 4. "The value of the comic book increased by an average of $[/tex]0.04 each month between [tex]\( m = 36 \)[/tex] and [tex]\( m = 60 \)[/tex]."
- This is incorrect because the rate of change is negative, indicating a decrease, not an increase.

Thus, the correct statement is:
- "The value of the comic book decreased by an average of $0.04 each month between [tex]\( m = 36 \)[/tex] and [tex]\( m = 60 \)[/tex]."
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