Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation `P = 108 − 23d`, where P is the number of phones left and d is the number of days she has worked that week. What is the meaning of the value 108 in this equation?

Options

A. Kathy will complete the repairs within 108 days.
B. Kathy starts each week with 108 phones to fix.
C. Kathy repairs phones at a rate of 108 per hour.
D. Kathy repairs phones at a rate of 108 per day.

Answer

To determine the meaning of the value 108 in the equation \( P = 108 - 23d \), we can analyze the equation step by step.

### Step 1: Understand the Variables
In the equation:
- \( P \) represents the number of phones left to repair.
- \( d \) represents the number of days Kathy has worked that week.

### Step 2: Substitute \( d = 0 \)
To find out what 108 represents, we can substitute \( d = 0 \) into the equation. This will give us the value of \( P \) when Kathy has not worked any days yet.

\[ P = 108 - 23(0) \]

### Step 3: Simplify the Equation
Now, simplify the equation:

\[ P = 108 - 0 \]
\[ P = 108 \]

### Step 4: Interpret the Result
When \( d = 0 \), the value of \( P \) is 108. This means that at the beginning of the week, before Kathy has started any repairs (0 days worked), she has 108 phones that need to be repaired.

### Conclusion
Thus, the value 108 represents the initial number of phones that Kathy has to repair at the beginning of the week when she hasn't started working yet.

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