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Answer :
To find the maximum height of a projectile, you can use the given equation that models the path of the projectile: [tex]\( h(t) = -16t^2 + 48t + 190 \)[/tex].
Here is a step-by-step guide to finding the maximum height:
1. Identify the form of the equation: The equation given is of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Find the vertex of the parabola: The height function [tex]\( h(t) \)[/tex] is a parabolic equation, and the highest point on the parabola occurs at its vertex. The time at which the vertex occurs can be calculated using the formula [tex]\( t = -\frac{b}{2a} \)[/tex].
3. Calculate the time at maximum height:
[tex]\[ t = -\frac{48}{2 \times -16} = -\frac{48}{-32} = 1.5 \][/tex]
So, the projectile reaches its maximum height at [tex]\( t = 1.5 \)[/tex] seconds.
4. Substitute [tex]\( t \)[/tex] into the equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
[tex]\[
h(1.5) = -16(2.25) + 72 + 190
\][/tex]
[tex]\[
h(1.5) = -36 + 72 + 190
\][/tex]
[tex]\[
h(1.5) = 226
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
Here is a step-by-step guide to finding the maximum height:
1. Identify the form of the equation: The equation given is of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Find the vertex of the parabola: The height function [tex]\( h(t) \)[/tex] is a parabolic equation, and the highest point on the parabola occurs at its vertex. The time at which the vertex occurs can be calculated using the formula [tex]\( t = -\frac{b}{2a} \)[/tex].
3. Calculate the time at maximum height:
[tex]\[ t = -\frac{48}{2 \times -16} = -\frac{48}{-32} = 1.5 \][/tex]
So, the projectile reaches its maximum height at [tex]\( t = 1.5 \)[/tex] seconds.
4. Substitute [tex]\( t \)[/tex] into the equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
[tex]\[
h(1.5) = -16(2.25) + 72 + 190
\][/tex]
[tex]\[
h(1.5) = -36 + 72 + 190
\][/tex]
[tex]\[
h(1.5) = 226
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
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