$T(d)$ 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 as 10 − 4 T ( 10 ) − T ( 4 ) .
Set the average rate of change to 0: 6 T ( 10 ) − T ( 4 ) = 0 .
Solve for the relationship between T ( 10 ) and T ( 4 ) , finding 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
Understanding the Problem 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 based on this information.
Setting up the Equation 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. So we have: 10 − 4 T ( 10 ) − T ( 4 ) = 0
Solving for the Difference We can simplify the denominator: 6 T ( 10 ) − T ( 4 ) = 0 To solve for T ( 10 ) − T ( 4 ) , we multiply both sides of the equation by 6: T ( 10 ) − T ( 4 ) = 0 × 6 T ( 10 ) − T ( 4 ) = 0
Interpreting the Result From the equation T ( 10 ) − T ( 4 ) = 0 , we can deduce that: 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 we analyze the given statements:
The same number of tickets was sold on the fourth day and tenth day. - This is true because T ( 10 ) = T ( 4 ) .
No tickets were sold on the fourth day and tenth day. - This is not necessarily true. T ( 4 ) and T ( 10 ) could be any equal number, not just 0.
Fewer tickets were sold on the fourth day than on the tenth day. - This is false because T ( 4 ) = T ( 10 ) .
More tickets were sold on the fourth day than on the tenth day. - This is false because T ( 4 ) = T ( 10 ) .
Therefore, the statement that must be true is: The same number of tickets was sold on the fourth day and tenth day.
Final Answer The correct statement is: The same number of tickets was sold on the fourth day and tenth day.
Examples
Understanding rates of change is crucial in many real-world scenarios. For instance, imagine you're tracking the growth of a plant over time. If the average rate of change in height between week 2 and week 4 is zero, it means the plant's height was the same in both weeks. This concept applies to stock prices, population growth, and even the speed of a car. Recognizing when rates of change are zero helps identify stable periods or equilibrium points in various dynamic systems.
The average rate of change of the function T ( d ) between days 4 and 10 being 0 implies that T ( 10 ) = T ( 4 ) . Therefore, the statement that must be true is that the same number of tickets was sold on the fourth day and tenth day. The correct answer is option A.
;
