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Answer :
To determine the maximum height of the projectile, we use the given equation for the height of the projectile over time:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is in the form of a quadratic function [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
The maximum height of a projectile, which follows a parabolic path, is reached at the vertex of the parabola. For a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], the time [tex]\( t \)[/tex] at which the maximum height occurs is calculated by the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values:
[tex]\[ t = -\frac{48}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
At [tex]\( t = 1.5 \)[/tex] seconds, the projectile reaches its maximum height. Now, we substitute [tex]\( t = 1.5 \)[/tex] into the height equation to find this maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculating step-by-step:
1. [tex]\( (1.5)^2 = 2.25 \)[/tex]
2. [tex]\( -16 \times 2.25 = -36 \)[/tex]
3. [tex]\( 48 \times 1.5 = 72 \)[/tex]
4. Put it all together:
[tex]\[ h(1.5) = -36 + 72 + 190 \][/tex]
5. [tex]\( -36 + 72 = 36 \)[/tex]
6. [tex]\( 36 + 190 = 226 \)[/tex]
So, the maximum height reached by the projectile is 226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is in the form of a quadratic function [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
The maximum height of a projectile, which follows a parabolic path, is reached at the vertex of the parabola. For a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], the time [tex]\( t \)[/tex] at which the maximum height occurs is calculated by the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values:
[tex]\[ t = -\frac{48}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
At [tex]\( t = 1.5 \)[/tex] seconds, the projectile reaches its maximum height. Now, we substitute [tex]\( t = 1.5 \)[/tex] into the height equation to find this maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculating step-by-step:
1. [tex]\( (1.5)^2 = 2.25 \)[/tex]
2. [tex]\( -16 \times 2.25 = -36 \)[/tex]
3. [tex]\( 48 \times 1.5 = 72 \)[/tex]
4. Put it all together:
[tex]\[ h(1.5) = -36 + 72 + 190 \][/tex]
5. [tex]\( -36 + 72 = 36 \)[/tex]
6. [tex]\( 36 + 190 = 226 \)[/tex]
So, the maximum height reached by the projectile is 226 feet.
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