To find the range of temperatures that ensures at least 50% of the kernels will pop, we need to set up the inequality based on the given equation and solve for $T$.

Step 1: Set Up the Inequality

The equation modeling the percent of kernels that pop is:

$P = -\frac{1}{250}T^2 + 2.8T - 394$

We want at least 50% of the kernels to pop, so we set up the inequality:

$-\frac{1}{250}T^2 + 2.8T - 394 \geq 50$

Step 2: Rearrange the Inequality

To make this a standard quadratic inequality, we subtract 50 from both sides:

$-\frac{1}{250}T^2 + 2.8T - 444 \geq 0$

Step 3: Multiply by -250 to Eliminate the Fraction

To simplify, multiply the entire inequality by $-250$ (note that multiplying by a negative number will reverse the inequality):

$T^2 - 700T + 111000 \leq 0$

Step 4: Solve the Quadratic Equation

Now, solve the quadratic equation $T^2 - 700T + 111000 = 0$ using the quadratic formula:

$T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 1$, $b = -700$, and $c = 111000$.

Step 5: Calculate the Discriminant

First, calculate the discriminant:

$\text{Discriminant} = b^2 - 4ac = (-700)^2 - 4(1)(111000)$

$\text{Discriminant} = 490000 - 444000 = 46000$

Step 6: Apply the Quadratic Formula

Now, solve for $T$:

$T = \frac{700 \pm \sqrt{46000}}{2}$

$T = \frac{700 \pm 214.5}{2}$

This gives us two solutions:

$T_1 = \frac{700 + 214.5}{2} = \frac{914.5}{2} = 457.3$

$T_2 = \frac{700 - 214.5}{2} = \frac{485.5}{2} = 242.8$

Step 7: Interpret the Solution

The temperatures $T$ that satisfy the original inequality are those where the quadratic expression is less than or equal to zero. The solution to the inequality is therefore:

$242.8 \leq T \leq 457.3$

Final Answer

The range of temperatures that ensures at least 50% of the kernels will pop is approximately 242.8°F to 457.3°F.

Would you like to go over the details or have any questions? Here are five follow-up questions you might consider:

1. How does the quadratic formula work, and why is the discriminant important?
2. What happens if the discriminant in a quadratic equation is negative?
3. How would the inequality change if we wanted exactly 50% of the kernels to pop?
4. How do the coefficients of the quadratic equation affect the shape of the parabola?
5. What is the significance of the temperature range calculated in a real-world context?

Tip: When working with inequalities involving quadratic equations, always remember that the direction of the inequality can change if you multiply or divide by a negative number.