Answered

[tex]$T(d)$[/tex] 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 [tex]$T(d)$[/tex] for the interval [tex]$d=4$[/tex] and [tex]$d=10$[/tex] is 0. Which statement must be true?

A. The same number of tickets was sold on the fourth day and the tenth day.
B. No tickets were sold on the fourth day and the 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 :

To determine which statement is true, we need to analyze the given information about the average rate of change of the function [tex]\( T(d) \)[/tex], which denotes the number of tickets sold on day [tex]\( d \)[/tex].

The average rate of change of [tex]\( T(d) \)[/tex] over the interval from [tex]\( d = 4 \)[/tex] to [tex]\( d = 10 \)[/tex] is given as 0. The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] can be calculated using the formula:
[tex]\[ \frac{T(b) - T(a)}{b - a} \][/tex]

In this case:
[tex]\[ a = 4 \quad \text{and} \quad b = 10 \][/tex]

Thus, the average rate of change is:
[tex]\[ \frac{T(10) - T(4)}{10 - 4} \][/tex]

Since we know that the average rate of change is 0, we can set up the following equation:
[tex]\[ \frac{T(10) - T(4)}{6} = 0 \][/tex]

To solve this equation, multiply both sides by 6 (the length of the interval):
[tex]\[ T(10) - T(4) = 0 \][/tex]

This simplifies to:
[tex]\[ T(10) = T(4) \][/tex]

This means that the number of tickets sold on the tenth day, [tex]\( T(10) \)[/tex], is equal to the number of tickets sold on the fourth day, [tex]\( T(4) \)[/tex].

Therefore, the correct statement is:
"The same number of tickets was sold on the fourth day and tenth day."