a. To determine the projectile's maximum height, we need to find the vertex of the quadratic function h(t) = -16t^2 + 272t.
The vertex of a quadratic function in the form of f(x) = ax^2 + bx + c is given by the formula:
x = -b / (2a)
In this case, a = -16 and b = 272, so substituting these values into the formula, we have:
t = -272 / (2 * -16)
Simplifying further:
t = -272 / -32
t = 8.5
Therefore, the projectile reaches its maximum height at t = 8.5 seconds.
To find the maximum height, substitute t = 8.5 into the function h(t):
h(8.5) = -16(8.5)^2 + 272(8.5)
Simplifying:
h(8.5) = -16(72.25) + 2312
h(8.5) = -1156 + 2312
h(8.5) = 1156
Therefore, the projectile's maximum height is 1156 feet.
b. We already determined that the projectile reaches its maximum height at t = 8.5 seconds.
c. To determine how long it will take for the projectile to strike the ground, we set h(t) = 0 (since the height above the ground is zero when the projectile strikes the ground).
So, we have:
0 = -16t^2 + 272t
Divide the equation by 8 to simplify it:
0 = -2t^2 + 34t
Now we can factor out a common factor:
0 = t(-2t + 34)
Setting each factor equal to zero, we have:
t = 0 or -2t + 34 = 0
Since time cannot be negative, we disregard t = 0.
Solving -2t + 34 = 0:
-2t = -34
Dividing by -2:
t = 17
Therefore, it will take the projectile 17 seconds to strike the ground after it is thrown.