To find the interval of selling prices that will allow Jazz to cover her monthly expenses of $4,000, we need to solve the inequality:
\[ y = -22.64x^2 + 789.35x - 2804.54 \geq 4000 \]
First, we rearrange the inequality:
\[ -22.64x^2 + 789.35x - 2804.54 - 4000 \geq 0 \]
This simplifies to:
\[ -22.64x^2 + 789.35x - 6804.54 \geq 0 \]
Next, we can multiply the entire inequality by -1 (which reverses the inequality):
\[ 22.64x^2 - 789.35x + 6804.54 \leq 0 \]
Now, we need to find the roots of the quadratic equation:
\[ 22.64x^2 - 789.35x + 6804.54 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 22.64\), \(b = -789.35\), and \(c = 6804.54\):
- Calculate the discriminant:
\[ D = b^2 - 4ac = (-789.35)^2 - 4(22.64)(6804.54) \]
\[ D \approx 622,495.6225 - 615,162.4864 \approx 6,333.1361 \]
- Calculate the roots:
\[ x = \frac{789.35 \pm \sqrt{6333.1361}}{2 \cdot 22.64} \]
\[ \sqrt{6333.1361} \approx 79.6 \]
Now, substituting in for \(x\):
\[ x_1 = \frac{789.35 + 79.6}{45.28} \approx \frac{868.95}{45.28} \approx 19.22 \] \[ x_2 = \frac{789.35 - 79.6}{45.28} \approx \frac{709.75}{45.28} \approx 15.70 \]
So, the roots of the quadratic are approximately \(x_1 \approx 19.22\) and \(x_2 \approx 15.70\). Since the parabola opens downward (the coefficient of \(x^2\) is negative), the values of \(x\) for which the profit is greater than or equal to $4,000 are between the two roots:
Thus, the interval of selling prices for T-shirts that will allow Jazz to cover all her business's monthly expenses is:
\[ \text{between } $15.70 \text{ and } $19.22 \]
Therefore, Jazz should set her T-shirt selling prices between $15.70 and $19.22.