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?
A. The value of the comic book decreased by a total of $0.04 between $m=36$ and $m=60$.
B. The value of the comic book increased by a total of $0.04 between $m=36$ and $m=60$.
C. The value of the comic book decreased by an average of $0.04 each month between $m=36$ and $m=60$.
D. The value of the comic book increased by an average of $0.04 each month between $m=36$ and $m=60$.
Answer (2)
The average rate of change of -0.04 indicates that the value of the comic book decreased by an average of 0.04 each month from month 36 to month 60. Thus, the correct answer is C. This means the comic book's value decreases steadily over the given time period.
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Calculate the difference in months: 60 − 36 = 24 .
Calculate the total change in value: − 0.04 × 24 = − $0.96 .
Interpret the average rate of change: The value decreased by an average of $0.04 each month.
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 decreased by an average of $0.04 each month between m = 36 and m = 60. .
Explanation
Understanding the Problem We are given that the average rate of change of the value, V ( m ) , of a comic book m months after publication is -0.04 between m = 36 and m = 60 . This means that on average, the value of the comic book decreases by $0.04 each month during this period. We need to determine which of the given statements must be true.
Setting up the Equation The average rate of change is calculated as the change in value divided by the change in time (in months). In this case, the average rate of change is given by: 60 − 36 V ( 60 ) − V ( 36 ) = − 0.04
Calculating the Time Difference First, let's calculate the difference in months: 60 − 36 = 24 months
Calculating the Total Change in Value Now, let's calculate the total change in value between m = 36 and m = 60 . We can rearrange the equation to find the total change in value: V ( 60 ) − V ( 36 ) = − 0.04 × 24 = − 0.96 This means that the value of the comic book decreased by a total of 0.96 b e tw ee n m=36 an d m=60$.
Determining the Correct Statement The average rate of change is -0.04, which means the value decreased by an average of 0.04 e a c hm o n t hb e tw ee n m=36 an d m=60$. Therefore, the statement 'The value of the comic book decreased by an average of 0.04 e a c hm o n t hb e tw ee n m=36 an d m=60$' must be true.
Final Answer The correct statement is: The value of the comic book decreased by an average of $0.04 each month between m = 36 and m = 60 .
Examples
Understanding rates of change is crucial in many real-world scenarios. For example, when analyzing the depreciation of a car's value over time, the average rate of change helps predict its resale value. Similarly, in economics, it can be used to model the inflation rate or the growth rate of a company's revenue. By calculating the average rate of change, we can make informed decisions about investments, budgeting, and financial planning. For instance, if a car's value depreciates at an average rate of $200 per month, we can estimate its value after a certain period by multiplying the rate by the number of months and subtracting it from the initial value. This concept is also applicable in scientific fields, such as calculating the rate of cooling of an object or the rate of decay of a radioactive substance.
