Answer :
Let's address the problem step by step to determine which statement must be true given that the average rate of change in the depth of the pool over the two-week interval is zero.
1. Understanding Average Rate of Change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is the change in the value of the function divided by the length of the interval. Mathematically, it is given by:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
- If this rate of change is zero, it means there is no net change in the function's value over the interval [tex]\( [a, b] \)[/tex].
2. Applying to Pool Depth:
- Let [tex]\(d(t)\)[/tex] be the depth of the pool at time [tex]\( t \)[/tex]. Over the interval from time [tex]\( t_1 \)[/tex] (start) to [tex]\( t_2 \)[/tex] (end), the depth did not change.
- The average rate of change is given by:
[tex]\[ \frac{d(t_2) - d(t_1)}{t_2 - t_1} \][/tex]
- Given that the average rate of change is zero, we have:
[tex]\[ \frac{d(t_2) - d(t_1)}{t_2 - t_1} = 0 \][/tex]
- This implies:
[tex]\[ d(t_2) - d(t_1) = 0 \][/tex]
[tex]\[ \text{Therefore, } d(t_2) = d(t_1) \][/tex]
3. Interpreting the Given Choices:
- Option 1: "The pool must have been empty for the entire interval."
- This statement is not necessarily true. The pool's depth could have been any constant value, not necessarily zero, as long as it remained unchanged.
- Option 2: "The pool must have been the same depth at the start of the interval as it was at the end of the interval."
- This statement accurately describes the condition we determined mathematically: [tex]\( d(t_2) = d(t_1) \)[/tex].
- Option 3: "The pool must have been deeper at the end of the interval than it was at the start of the interval."
- This statement is false since we determined the depth did not change.
- Option 4: "The pool must have been more shallow at the end of the interval than it was at the start of the interval."
- Similarly, this statement is false for the same reason.
4. Conclusion:
- The correct statement is:
[tex]\[ \text{The pool must have been the same depth at the start of the interval as it was at the end of the interval.} \][/tex]
So, the true statement is:
The pool must have been the same depth at the start of the interval as it was at the end of the interval.
1. Understanding Average Rate of Change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is the change in the value of the function divided by the length of the interval. Mathematically, it is given by:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
- If this rate of change is zero, it means there is no net change in the function's value over the interval [tex]\( [a, b] \)[/tex].
2. Applying to Pool Depth:
- Let [tex]\(d(t)\)[/tex] be the depth of the pool at time [tex]\( t \)[/tex]. Over the interval from time [tex]\( t_1 \)[/tex] (start) to [tex]\( t_2 \)[/tex] (end), the depth did not change.
- The average rate of change is given by:
[tex]\[ \frac{d(t_2) - d(t_1)}{t_2 - t_1} \][/tex]
- Given that the average rate of change is zero, we have:
[tex]\[ \frac{d(t_2) - d(t_1)}{t_2 - t_1} = 0 \][/tex]
- This implies:
[tex]\[ d(t_2) - d(t_1) = 0 \][/tex]
[tex]\[ \text{Therefore, } d(t_2) = d(t_1) \][/tex]
3. Interpreting the Given Choices:
- Option 1: "The pool must have been empty for the entire interval."
- This statement is not necessarily true. The pool's depth could have been any constant value, not necessarily zero, as long as it remained unchanged.
- Option 2: "The pool must have been the same depth at the start of the interval as it was at the end of the interval."
- This statement accurately describes the condition we determined mathematically: [tex]\( d(t_2) = d(t_1) \)[/tex].
- Option 3: "The pool must have been deeper at the end of the interval than it was at the start of the interval."
- This statement is false since we determined the depth did not change.
- Option 4: "The pool must have been more shallow at the end of the interval than it was at the start of the interval."
- Similarly, this statement is false for the same reason.
4. Conclusion:
- The correct statement is:
[tex]\[ \text{The pool must have been the same depth at the start of the interval as it was at the end of the interval.} \][/tex]
So, the true statement is:
The pool must have been the same depth at the start of the interval as it was at the end of the interval.
