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(t)=-16t^2+70t+3 where t is time in seconds and t>=0. Which is closest to the time when the softball will be 50 feet above the ground?
a. 0.66 seconds
b. 5.03 seconds
c. 0.83 seconds or 3.55 seconds
d. 0.66 seconds or 5.03 seconds
To find the time when the softball will be 50 feet above the ground, we need to set the height function equal to 50 and solve for t:
-16t^2 + 70t + 3 = 50
Simplifying this equation, we get:
-16t^2 + 70t - 47 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Since factoring is not straightforward in this case, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/2a
In this case, a = -16, b = 70, and c = -47. Plugging these values into the formula, we get:
t = (-70 ± √(70^2 - 4(-16)(-47)))/(2(-16))
t = (-70 ± √(4900 - 3016))/(-32)
t = (-70 ± √(1884))/(-32)
Simplifying further, we get:
t = (-70 ± √(9 * 4 * 47))/(-32)
t = (-70 ± √(3^2 * 2^2 * 47))/(-32)
t = (-70 ± 6√47)/(-32)
Now we can find the two possible values for t:
t1 = (-70 + 6√47)/(-32) ≈ 0.83 seconds
t2 = (-70 - 6√47)/(-32) ≈ 3.55 seconds
Therefore, the closest time when the softball will be 50 feet above the ground is either 0.83 seconds or 3.55 seconds.
Therefore, the correct answer is c. 0.83 seconds or 3.55 seconds.