Question

lionel computed the average rate of change in the depth of a pool over a two - week interval to be zero. which statement must be true?
the pool must have been empty for the entire interval.
the pool must have been the same depth at the start of the interval as it was at the end of the interval.
the pool must have been deeper at the end of the interval than it was at the start of the interval.
the pool must have been more shallow at the end of the interval than it was at the start of the interval.

Explanation:

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

In this problem, the "function" is the depth of the pool over time, and the interval is two weeks (so \( b - a \) is the length of the interval, which is non - zero). We are told that the average rate of change is zero.

Step 1: Set up the formula for average rate of change

Let \( f(a) \) be the depth of the pool at the start of the two - week interval (time \( a \)) and \( f(b) \) be the depth at the end of the interval (time \( b \)). The average rate of change of the depth is \( \frac{f(b)-f(a)}{b - a}=0 \).

Step 2: Analyze the equation

Since \( b - a
eq0 \) (because the interval is two weeks, a non - zero length of time), for the fraction \( \frac{f(b)-f(a)}{b - a} \) to be equal to zero, the numerator must be zero. That is, \( f(b)-f(a) = 0 \), which implies \( f(b)=f(a) \). This means the depth of the pool at the start of the interval is equal to the depth at the end of the interval.

Now let's analyze the other options:

• Option 1: The pool does not have to be empty for the entire interval. For example, if the depth starts at 5 feet, goes up to 10 feet, and then comes back down to 5 feet over the two - week period, the average rate of change is zero, but the pool was not empty.
• Option 3: If the pool was deeper at the end (\( f(b)>f(a) \)), then \( f(b)-f(a)>0 \), and the average rate of change would be positive, not zero.
• Option 4: If the pool was more shallow at the end (\( f(b)<f(a) \)), then \( f(b)-f(a)<0 \), and the average rate of change would be negative, not zero.

Answer:

The pool must have been the same depth at the start of the interval as it was at the end of the interval.