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Answer :
To find the maximum height of the projectile, we need to analyze the quadratic equation that describes its path:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation models the height [tex]\( h \)[/tex] of the projectile at time [tex]\( t \)[/tex]. Since it's a quadratic equation with a negative coefficient for [tex]\( t^2 \)[/tex], the parabola opens downward, indicating that the projectile will reach a maximum height at its vertex.
The formula for the time [tex]\( t \)[/tex] at which the maximum height occurs is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex] and [tex]\( b = 48 \)[/tex].
1. Determine the time to reach maximum height:
[tex]\[
t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5
\][/tex]
2. Find the maximum height:
Substitute [tex]\( t = 1.5 \)[/tex] back into the height equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
Calculate each component:
- [tex]\( -16 \times (1.5)^2 = -16 \times 2.25 = -36 \)[/tex]
- [tex]\( 48 \times 1.5 = 72 \)[/tex]
- Then add these results to the constant term:
[tex]\[
h(1.5) = -36 + 72 + 190 = 226
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation models the height [tex]\( h \)[/tex] of the projectile at time [tex]\( t \)[/tex]. Since it's a quadratic equation with a negative coefficient for [tex]\( t^2 \)[/tex], the parabola opens downward, indicating that the projectile will reach a maximum height at its vertex.
The formula for the time [tex]\( t \)[/tex] at which the maximum height occurs is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex] and [tex]\( b = 48 \)[/tex].
1. Determine the time to reach maximum height:
[tex]\[
t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5
\][/tex]
2. Find the maximum height:
Substitute [tex]\( t = 1.5 \)[/tex] back into the height equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
Calculate each component:
- [tex]\( -16 \times (1.5)^2 = -16 \times 2.25 = -36 \)[/tex]
- [tex]\( 48 \times 1.5 = 72 \)[/tex]
- Then add these results to the constant term:
[tex]\[
h(1.5) = -36 + 72 + 190 = 226
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
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