https://file.aiquickdraw.com/mathbot/file/1727050714084x8a605ax.jpg

To solve the problem, we need to find which of the given solutions result in a positive demand \( C(t) \) based on the function provided:

\[
C(t) = -\sqrt{t^2 + 2t - 15} + 3
\]

### Step 1: Analyze the function
The demand function \( C(t) \) involves a square root, and it's important to remember that the square root function is only defined for non-negative values. Therefore, \( t^2 + 2t - 15 \geq 0 \).

### Step 2: Find the roots of the quadratic equation
We need to solve the inequality:

\[
t^2 + 2t - 15 = 0
\]

To find the roots, use the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 2 \), and \( c = -15 \). Plugging these values into the formula gives:

\[
t = \frac{-2 \pm \sqrt{2^2 - 4(1)(-15)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 60}}{2} = \frac{-2 \pm \sqrt{64}}{2} = \frac{-2 \pm 8}{2}
\]

This results in:

\[
t = \frac{-2 + 8}{2} = 3 \quad \text{and} \quad t = \frac{-2 - 8}{2} = -5
\]

### Step 3: Determine the sign of \( C(t) \)
- For \( t > 3 \) or \( t < -5 \), \( t^2 + 2t - 15 \) is positive, making the square root term valid, but \( C(t) \) becomes negative since it's \( -\sqrt{\text{positive value}} + 3 \).
- For \( -5 \leq t \leq 3 \), the square root term will be non-positive. Therefore, to get a positive \( C(t) \), the square root term must be less than 3.

### Step 4: Check the given options
We check each \( t \) value:

1. **For \( t = 5 \):**
   \[
   C(5) = -\sqrt{5^2 + 2(5) - 15} + 3 = -\sqrt{25 + 10 - 15} + 3 = -\sqrt{20} + 3 \quad (\text{negative value})
   \]

2. **For \( t = 3 \):**
   \[
   C(3) = -\sqrt{3^2 + 2(3) - 15} + 3 = -\sqrt{9 + 6 - 15} + 3 = -\sqrt{0} + 3 = 3 \quad (\text{positive value})
   \]

3. **For \( t = 0 \):**
   \[
   C(0) = -\sqrt{0^2 + 2(0) - 15} + 3 = -\sqrt{-15} + 3 \quad (\text{undefined since square root of negative})
   \]

Thus, the only valid solution with a positive demand is at \( t = 3 \).

### Answer
The correct choice is \( (3, 3) \), which corresponds to the second option.

---

Would you like any further explanation, or do you have any other questions?

Here are five related questions you might find useful:
1. What is the importance of determining the domain of a function?
2. How do you find the vertex of a quadratic function?
3. How do you determine if a quadratic function opens upwards or downwards?
4. Can a quadratic function have two positive roots?
5. How does the discriminant affect the nature of the roots of a quadratic equation?

**Tip:** Always check the domain of a function when dealing with square roots or logarithms, as these operations have restrictions on the values they can take.

When a new cellphone is put on the market, the demand each month can be described by the function C(t) = -√(t^2 + 2t - 15) + 3, where C(t) represents the demand of the cellphone (measured in millions of people) and t is measured in months. Which of the following solution(s) are valid for a positive demand?

Learn how to determine the values of t for which the demand function C(t) = -√(t^2 + 2t - 15) + 3 is positive. This problem explores quadratic equations, square root functions, and inequalities. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation