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Answer :
To find the maximum height of the projectile, we can use the model given by the equation:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a quadratic function in the form of [tex]\( h(t) = at^2 + bt + c \)[/tex] where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
For a quadratic equation, the maximum (or minimum) height is found at the vertex of the parabola. Since the coefficient of [tex]\( t^2 \)[/tex] is negative ([tex]\( a < 0 \)[/tex]), the parabola opens downward, and therefore the vertex represents the maximum point.
The formula to find the time [tex]\( t \)[/tex] at which the maximum height occurs is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
This means the maximum height occurs at [tex]\( t = 1.5 \)[/tex] seconds after launch.
Next, we calculate the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the original equation for [tex]\( h(t) \)[/tex]:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculating this:
[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.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a quadratic function in the form of [tex]\( h(t) = at^2 + bt + c \)[/tex] where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
For a quadratic equation, the maximum (or minimum) height is found at the vertex of the parabola. Since the coefficient of [tex]\( t^2 \)[/tex] is negative ([tex]\( a < 0 \)[/tex]), the parabola opens downward, and therefore the vertex represents the maximum point.
The formula to find the time [tex]\( t \)[/tex] at which the maximum height occurs is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
This means the maximum height occurs at [tex]\( t = 1.5 \)[/tex] seconds after launch.
Next, we calculate the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the original equation for [tex]\( h(t) \)[/tex]:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculating this:
[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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