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Answer :
To find the maximum height of the projectile, we need to analyze the path of the projectile which is modeled by the height equation:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This is a quadratic equation where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
In mathematics, the maximum height of a projectile (when modeled by a quadratic equation like this) occurs at the vertex of the parabola. The formula to find the time [tex]\( t \)[/tex] when the maximum height is reached for a parabola described by [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values from the equation:
1. Calculate the time [tex]\( t \)[/tex] at which max height is reached:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
2. 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]
3. Compute the value:
[tex]\[ h(1.5) = -16 \times 2.25 + 72 + 190 \][/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.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This is a quadratic equation where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
In mathematics, the maximum height of a projectile (when modeled by a quadratic equation like this) occurs at the vertex of the parabola. The formula to find the time [tex]\( t \)[/tex] when the maximum height is reached for a parabola described by [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values from the equation:
1. Calculate the time [tex]\( t \)[/tex] at which max height is reached:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
2. 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]
3. Compute the value:
[tex]\[ h(1.5) = -16 \times 2.25 + 72 + 190 \][/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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