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?
To determine which statement must be true, we need to understand what the average rate of change represents and how it can be calculated.
The average rate of change in a function over an interval can be found using the formula:
Average Rate of Change = (Change in Function Values) / (Change in Input Values)
In this case, the function is T(d), which represents the number of tickets sold for a movie, and the input values are the number of days since the movie was released (d).
Given that the average rate of change in T(d) for the interval d = 4 and d = 10 is 0, we can write this information as:
Average Rate of Change = 0
Now, let's consider the possible statements and determine which one must be true:
1. T(10) - T(4) = 0
2. T(10) = T(4)
3. T(4) = 0
4. T(10) = 0
To find the correct statement, we need to use the formula for the average rate of change mentioned earlier. Since the average rate of change is 0, it implies that the change in the function values is 0 over the given interval.
Therefore, the correct statement is:
2. T(10) = T(4)
This statement means that the number of tickets sold for the movie at day 10 is equal to the number of tickets sold at day 4.
To find the average rate of change of T(d) for the interval between d = 4 and d = 10, we need to calculate the difference in T(d) values and divide it by the difference in d values.
Let's calculate it step by step:
1. Subtract the initial value of T(d) from the final value of T(d) within the given interval:
T(10) - T(4)
2. Divide the difference in T(d) values by the difference in d values:
(T(10) - T(4)) / (10 - 4)
If the average rate of change in T(d) for this interval is 0, then the above expression must equal 0:
(T(10) - T(4)) / (10 - 4) = 0
However, without additional information about the function T(d) or the specific values of T(10) and T(4), we cannot determine the exact statement that must be true.