To determine the maximum height of the projectile, we need to analyze the quadratic function given: [tex]\( h(t) = -16t^2 + 400t \)[/tex]. This function represents the height of the projectile at any given time [tex]\( t \)[/tex].
### Step-by-step Solution:
1. Identify the Coefficients:
The quadratic function is in the standard form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 400 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Determine the Vertex:
The maximum height of a projectile in a parabolic path occurs at the vertex of the parabola. For a parabola represented by [tex]\( at^2 + bt + c \)[/tex], the time [tex]\( t \)[/tex] at which the maximum height occurs is found using the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
Plugging in the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
t = -\frac{400}{2 \cdot -16} = \frac{400}{32} = 12.5
\][/tex]
Thus, the time at which the projectile reaches its maximum height is [tex]\( t = 12.5 \)[/tex] seconds.
3. Calculate the Maximum Height:
Substitute [tex]\( t = 12.5 \)[/tex] back into the height function to find the maximum height:
[tex]\[
h(12.5) = -16 (12.5)^2 + 400(12.5)
\][/tex]
Simplify inside the parentheses:
[tex]\[
(12.5)^2 = 156.25
\][/tex]
Then multiply:
[tex]\[
-16 \cdot 156.25 = -2500
\][/tex]
And:
[tex]\[
400 \cdot 12.5 = 5000
\][/tex]
Adding these together:
[tex]\[
h(12.5) = -2500 + 5000 = 2500
\][/tex]
Hence, the maximum height of the projectile is [tex]\( 2500 \)[/tex] feet.
### Conclusion:
The maximum height of the object is:
2500 feet
