Sandra bought a new computer for $600 and is paying on an installment plan. She paid $100 at the time of purchase and will make $50 monthly payments until the computer is paid off. The function C(p) represents the amount Sandra owes on the computer after making p payments of $50.
What is the domain of C(p)?

To determine the domain of $ C(p) $, the function representing the amount Sandra owes on the computer after making $ p $ payments of $50, we need to understand how the function is defined and what $ p $ represents.

Given:
- Sandra bought a computer for $600.
- She paid $100 at the time of purchase.
- The remaining balance is $600 - $100 = $500.
- She will make monthly payments of $50 until the computer is paid off.

The function $ C(p) $ describes the remaining balance after $ p $ payments of $50. Thus, $ C(p) $ can be expressed as:
\[ C(p) = 500 - 50p \]

The next step is to find the range of $ p $ for which the balance remains non-negative. The amount Sandra owes should not be negative, so:
\[ 500 - 50p \geq 0 \]

Solving for $ p $:
\[ 500 \geq 50p \]
\[ 10 \geq p \]
\[ p \leq 10 \]

Since $ p $ represents the number of payments and she cannot make a negative number of payments, $ p $ must also be non-negative:
\[ p \geq 0 \]

Combining these conditions, the domain of $ C(p) $ is:
\[ 0 \leq p \leq 10 \]

Therefore, the domain of $ C(p) $ is:
\[ \{ p \in \mathbb{R} \mid 0 \leq p \leq 10 \} \]

Would you like more details on this solution or have any questions? 

Here are some related questions to explore:

1. What is the range of the function $ C(p) $?
2. How many payments will Sandra have made when she owes $200 on the computer?
3. What will be the remaining balance after 5 payments?
4. If Sandra doubles her monthly payment, how does the domain of $ C(p) $ change?
5. What is the function $ C(p) $ if the initial payment was $200 instead of $100?
6. How would the total amount paid change if Sandra paid the entire amount upfront?
7. What is the total amount Sandra pays including the initial payment and all monthly installments?
8. How would the function $ C(p) $ change if Sandra missed a payment?

**Tip:** To determine the domain of a function, always consider the constraints and the practical context in which the function operates.

Explore the domain of the function C(p) in Sandra's installment payment problem where she buys a computer for $600, pays $100 upfront, and makes monthly $50 payments. Understand the constraints and conditions for the number of payments p. Suitable for Grades 7-9.

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