Question

The table above shows the numbers of hours of television programs that Jane recorded last week and the numbers of hours she spent viewing these recorded programs. No recorded program was viewed more than once. If h is the number of hours of recorded programs she had not yet viewed by the end of Friday, which of the following intervals represents all of the possible values of h?

• A. \(0 \le h \le 1\)
• B. \(1 \le h \le 2\)
• C. \(2 \le h \le 3\)
• D. \(0 \le h \le 2\)
• E. \(1 \le h \le 3\)

Hint:

To find the range of a difference (like \(A - B\)), the logic is:
\( \text{Max Difference} = A_{max} - B_{min} \)
\( \text{Min Difference} = A_{min} - B_{max} \)
In this problem, the recorded time was a fixed value (so \(A_{max} = A_{min} = 6\)), which simplifies the calculation.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem requires calculating a range of possible values. The number of unviewed hours (\(h\)) is the difference between the total hours recorded and the total hours viewed. Since the viewing time is given as a range, the unviewed time will also be a range.
Step 2: Key Formula or Approach:
Unviewed Hours (h) = Total Recorded Hours - Total Viewed Hours
We need to find the minimum and maximum possible values for \(h\).
Step 3: Detailed Explanation:
1. Calculate the Total Recorded Hours.
From the table, Jane recorded for 4 hours on Tuesday and 2 hours on Thursday.
\[ \text{Total Recorded Hours} = 4 + 2 = 6 \text{ hours} \] 2. Calculate the Range of Total Viewed Hours.
Let \(V\) be the total viewing time.
Viewing on Wednesday: \(1 \le V_{Wed} \le 2\) hours.
Viewing on Friday: \(2 \le V_{Fri} \le 3\) hours.
The total viewing time is the sum of these two ranges.
Minimum Total Viewed Hours = (Min Wed) + (Min Fri) = \(1 + 2 = 3\) hours.
Maximum Total Viewed Hours = (Max Wed) + (Max Fri) = \(2 + 3 = 5\) hours.
So, the range for the total viewed hours is \(3 \le V \le 5\).
3. Calculate the Range for Unviewed Hours (h).
\[ h = 6 - V \] To find the minimum value of \(h\), we must subtract the maximum possible value of \(V\).
\[ h_{min} = 6 - V_{max} = 6 - 5 = 1 \] To find the maximum value of \(h\), we must subtract the minimum possible value of \(V\).
\[ h_{max} = 6 - V_{min} = 6 - 3 = 3 \] Therefore, the interval representing all possible values of \(h\) is \(1 \le h \le 3\).
Step 4: Final Answer:
The interval for \(h\) is from 1 to 3, inclusive. This corresponds to option (E).