Final answer:
The football spends a total of 3.25 seconds at or above 67 feet. This is found by solving the quadratic equation for the time when the football is at 67 feet, resulting in two-time values, 3.79 seconds and 0.54 seconds. The difference between these times gives the duration above 67 feet.
Step-by-step explanation:
To calculate the time the football spends at or above a height of 67 feet, we need to find when the height (y) equals 67 feet using the given quadratic equation: y = -16x2 + 80x + 3. We set this equation equal to 67 and solve for x using the quadratic formula. Here's the adjusted equation:
67 = -16x2 + 80x + 3.
First, we rearrange the equation: 0 = -16x2 + 80x - 64. Now we apply the quadratic formula x = (-b ± √(b2 - 4ac)) / (2a), where a = -16, b = 80, and c = -64. Using these values, we calculate the two possible solutions for x.
After solving, we find two-time values that correspond to when the ball is at 67 feet: t = 3.79 s and t = 0.54 s. To find the total time the ball spends at or above 67 feet, we simply subtract the smaller time from the larger time:
Total time at or above 67 feet = 3.79 s - 0.54 s = 3.25 seconds.