To determine the time interval during which the projectile is at least 125 feet above the ground, we need to analyze the given quadratic equation: y = -2x² + 38x + 10. Here, y represents the height in feet, and x represents time in seconds. Our goal is to find the values of x for which y is greater than or equal to 125 feet.
Setting Up the Inequality
First, we set up the inequality based on the height requirement:
-2x² + 38x + 10 ≥ 125
Rearranging the Inequality
Next, we rearrange this inequality to bring all terms to one side:
-2x² + 38x + 10 - 125 ≥ 0
This simplifies to:
-2x² + 38x - 115 ≥ 0
Factoring the Quadratic
To solve the quadratic inequality, we first need to find the roots of the corresponding equation:
-2x² + 38x - 115 = 0
We can use the quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, where a = -2, b = 38, and c = -115.
Calculating the Discriminant
First, we calculate the discriminant (b² - 4ac):
- b² = 38² = 1444
- 4ac = 4 * (-2) * (-115) = 920
- Discriminant = 1444 - 920 = 524
Finding the Roots
Now, we can find the roots:
x = (−38 ± √524) / (2 * -2)
Calculating √524 gives approximately 22.9, so:
x = (−38 ± 22.9) / -4
This results in two potential solutions:
- x₁ = (−38 + 22.9) / -4 ≈ 3.78
- x₂ = (−38 - 22.9) / -4 ≈ 15.78
Analyzing the Interval
Now that we have the roots, we need to determine the intervals where the quadratic expression is greater than or equal to zero. The roots divide the number line into three intervals:
- (-∞, 3.78)
- (3.78, 15.78)
- (15.78, ∞)
To find where the expression is non-negative, we can test points from each interval:
- For x = 0 (in the first interval): -2(0)² + 38(0) - 115 = -115 (negative)
- For x = 5 (in the second interval): -2(5)² + 38(5) - 115 = 25 (positive)
- For x = 20 (in the third interval): -2(20)² + 38(20) - 115 = -115 (negative)
Conclusion on the Time Interval
From our testing, we see that the quadratic expression is non-negative in the interval [3.78, 15.78]. Therefore, the projectile is at least 125 feet above the ground during the time interval:
[3.8 seconds, 15.8 seconds]
Thus, rounding to the nearest tenth of a second, the projectile is at least 125 feet above the ground from approximately 3.8 seconds to 15.8 seconds.
