During a softball game, a batter hits a ball upward from an initial height of 3 feet. The height, in feet, of the softball is given by s left-parenthesis t right-parenthesis equals negative 16 t squared plus 70 t plus 3, where t is time in seconds and t greater than or equal to 0. Which is closest to the time when the softball will be 50 feet above the ground?
To find the time when the softball will be 50 feet above the ground, we need to set the height equation equal to 50 and solve for t.
-16t^2 + 70t + 3 = 50
Subtracting 50 from both sides:
-16t^2 + 70t - 47 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -16, b = 70, and c = -47.
t = (-70 ± √(70^2 - 4*(-16)*(-47))) / 2*(-16)
t = (-70 ± √(4900 - 3016)) / (-32)
t = (-70 ± √(1884)) / (-32)
Calculating the square root of 1884, we get:
t ≈ (-70 ± 43.39) / (-32)
Now we have two solutions:
1) t = (-70 + 43.39) / (-32) ≈ (-26.61) / (-32) ≈ 0.831 seconds
2) t = (-70 - 43.39) / (-32) ≈ (-113.39) / (-32) ≈ 3.543 seconds
Therefore, the time when the softball will be closest to 50 feet above the ground is approximately 0.831 seconds.