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Answer :
Sure, let's find the maximum height of the projectile step-by-step.
We are given the height equation for the projectile:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This is a quadratic equation in the standard form:
[tex]\[ h(t) = at^2 + bt + c \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
To find the maximum height of the projectile, we need to find the vertex of this parabola. The vertex of a parabola described by the equation [tex]\( at^2 + bt + c \)[/tex] occurs at [tex]\( t = -\frac{b}{2a} \)[/tex].
Step 1: Find the time at which the maximum height occurs.
[tex]\[ t = -\frac{b}{2a} \][/tex]
[tex]\[ t = -\frac{48}{2(-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, the maximum height occurs at [tex]\( t = 1.5 \)[/tex] seconds.
Step 2: Calculate the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the height equation.
[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.
So, the correct answer is:
[tex]\[ 226 \text{ feet} \][/tex]
We are given the height equation for the projectile:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This is a quadratic equation in the standard form:
[tex]\[ h(t) = at^2 + bt + c \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
To find the maximum height of the projectile, we need to find the vertex of this parabola. The vertex of a parabola described by the equation [tex]\( at^2 + bt + c \)[/tex] occurs at [tex]\( t = -\frac{b}{2a} \)[/tex].
Step 1: Find the time at which the maximum height occurs.
[tex]\[ t = -\frac{b}{2a} \][/tex]
[tex]\[ t = -\frac{48}{2(-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, the maximum height occurs at [tex]\( t = 1.5 \)[/tex] seconds.
Step 2: Calculate the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the height equation.
[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.
So, the correct answer is:
[tex]\[ 226 \text{ feet} \][/tex]
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