A particle is moving along a projectile path at an initial height of 96 feet with an initial speed of 16 feet per second. This can be represented by the function H(t) = −16t2 + 16t + 96. What is the maximum height of the particle?
100 feet
96 feet
60 feet
56 feet
The maximum height occurs at the vertex of the parabolic function, which is given by t=-b/2a. In this case, a=-16 and b=16, so t=-16/(2*(-16))=0.5 seconds. Plugging this time into the function, we get H(0.5)=-16(0.5)^2 + 16(0.5) + 96 = 100 feet. Therefore, the maximum height of the particle is 100 feet. Answer: \boxed{100\text{ feet}}.
To find the maximum height of the particle, we need to determine the vertex of the quadratic function H(t) = -16t^2 + 16t + 96.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).
In this case, a = -16 and b = 16. Plugging in these values into the formula, we get:
t = -16 / (2 * -16) = 0.5
To find the maximum height, we substitute this value of t back into the function:
H(0.5) = -16(0.5)^2 + 16(0.5) + 96
= -16(0.25) + 8 + 96
= -4 + 8 + 96
= 100
Therefore, the maximum height of the particle is 100 feet.