T(d) is a function that relates the number of tickets sold for a movie to the number of days since the movie was released.
The average rate of change in T(d) for the interval d = 4 and d = 10 is 0. Which statement must be true?
A. The same number of tickets was sold on the fourth day and tenth day.
B. No tickets were sold on the fourth day and tenth day.
C. Fewer tickets were sold on the fourth day than on the tenth day.
D. More tickets were sold on the fourth day than on the tenth day.
Answer (2)
Calculate the average rate of change between d=4 and d=10: 10 − 4 T ( 10 ) − T ( 4 ) = 0 .
Simplify the equation: T ( 10 ) − T ( 4 ) = 0 .
Solve for the relationship between T(10) and T(4): T ( 10 ) = T ( 4 ) .
Conclude that the same number of tickets was sold on the fourth and tenth days: The same number of tickets was sold on the fourth day and tenth day.
Explanation
Problem Analysis We are given that T ( d ) is a function that relates the number of tickets sold for a movie to the number of days since the movie was released. The average rate of change in T ( d ) for the interval d = 4 and d = 10 is 0. We need to determine which statement must be true.
Average Rate of Change Formula The average rate of change of a function T ( d ) between two points d 1 and d 2 is given by the formula: d 2 − d 1 T ( d 2 ) − T ( d 1 ) In our case, d 1 = 4 and d 2 = 10 , and the average rate of change is 0. Therefore, we have: 10 − 4 T ( 10 ) − T ( 4 ) = 0
Solving for the Relationship To solve for the relationship between T ( 10 ) and T ( 4 ) , we can multiply both sides of the equation by ( 10 − 4 ) , which is 6: T ( 10 ) − T ( 4 ) = 0 × 6 T ( 10 ) − T ( 4 ) = 0
Interpreting the Result Adding T ( 4 ) to both sides of the equation, we get: T ( 10 ) = T ( 4 ) This means that the number of tickets sold on day 10 is equal to the number of tickets sold on day 4.
Determining the Correct Statement Now, let's examine the given statements:
The same number of tickets was sold on the fourth day and tenth day.
No tickets were sold on the fourth day and tenth day.
Fewer tickets were sold on the fourth day than on the tenth day.
More tickets were sold on the fourth day than on the tenth day. Since T ( 10 ) = T ( 4 ) , the first statement, "The same number of tickets was sold on the fourth day and tenth day," must be true.
Final Answer Therefore, the correct answer is: The same number of tickets was sold on the fourth day and tenth day.
Examples
Understanding average rates of change is crucial in many real-world scenarios. For instance, consider a company tracking its revenue over time. If the average rate of change of revenue between two months is zero, it indicates that the company's revenue remained the same during that period. This concept is also applicable in fields like physics, where it can describe constant velocity, or in environmental science, where it can represent a stable population size.
The average rate of change of tickets sold between days 4 and 10 is zero, indicating that the number of tickets sold on both days is equal. Therefore, the correct choice is that the same number of tickets was sold on the fourth and tenth days. The answer is option A.
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