Question

is a projectile is fired straight upward from the ground with an initial speed of 224 feet per second, then its height h in feet after t seconds is given by the function h(t) = -16t² + 224t. find the maximum height of the projectile. (simplify your answer)

Explanation:

Step1: Identify the vertex formula for a quadratic function

The quadratic function is in the form \( h(t) = at^2 + bt + c \), and the \( t \)-coordinate of the vertex (which gives the time at maximum height) is \( t = -\frac{b}{2a} \). Here, \( a = -16 \) and \( b = 224 \).
\[
t = -\frac{224}{2\times(-16)}
\]

Step2: Calculate the time \( t \)

Simplify the expression for \( t \):
\[
t = -\frac{224}{-32} = 7
\]

Step3: Find the maximum height by substituting \( t = 7 \) into \( h(t) \)

Substitute \( t = 7 \) into \( h(t) = -16t^2 + 224t \):
\[
h(7) = -16\times(7)^2 + 224\times7
\]
\[
h(7) = -16\times49 + 1568
\]
\[
h(7) = -784 + 1568
\]
\[
h(7) = 784
\]

Answer:

784