Draco steals and throws Neville's red ball straight up at 86 feet per second from about 11 feet above the ground. How long does he have until the ball reaches the ground? HINT: Use the function h(t)=−16t2+86t+11, where h(t) is the height of the red ball after t seconds.
So, take their hint, set
−16t2+86t+11 = 0
16t^2 - 86t - 11 = 0
solve using the quadratic equation, reject the negative answer.
To find out how long it takes for the ball to reach the ground, we need to determine when its height becomes zero.
The equation provided h(t) = -16t^2 + 86t + 11 represents the height of the red ball over time, where t is the time in seconds.
To find out when the ball reaches the ground, we need to solve the equation h(t) = 0.
Therefore, we set -16t^2 + 86t + 11 = 0.
To solve this quadratic equation, we can either use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation -16t^2 + 86t + 11 = 0, the coefficients are:
a = -16, b = 86, and c = 11.
Using the quadratic formula, we can now calculate the values of t:
t = (-86 ± √(86^2 - 4(-16)(11))) / (2(-16))
Simplifying this equation, we have:
t = (-86 ± √(7396 + 704)) / (-32)
t = (-86 ± √(8100)) / (-32)
t = (-86 ± 90) / (-32)
Now we have two possible solutions for t:
t1 = (-86 + 90) / (-32)
t2 = (-86 - 90) / (-32)
Simplifying these expressions, we get:
t1 = 4 / (-32) = -1/8
t2 = -176 / (-32) = 11/2
Since time cannot be negative in this context, we disregard the negative value and consider the positive value.
So, Draco has approximately 11/2 or 5.5 seconds until the ball reaches the ground.