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Answer :
To find the maximum height of the projectile, we can use the given quadratic equation for its path:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a parabola that opens downwards because the coefficient of [tex]\( t^2 \)[/tex] is negative. The maximum height will occur at the vertex of this parabola.
### Step-by-Step Solution:
1. Identify the coefficients:
- The quadratic equation is in the form of [tex]\( at^2 + bt + c \)[/tex].
- Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Determine the time at which maximum height occurs:
- The time [tex]\( t \)[/tex] at which the vertex (maximum point) occurs can be calculated using the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
- Substitute the coefficients into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = 1.5 \][/tex]
3. Calculate 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]
[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 parabola that opens downwards because the coefficient of [tex]\( t^2 \)[/tex] is negative. The maximum height will occur at the vertex of this parabola.
### Step-by-Step Solution:
1. Identify the coefficients:
- The quadratic equation is in the form of [tex]\( at^2 + bt + c \)[/tex].
- Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Determine the time at which maximum height occurs:
- The time [tex]\( t \)[/tex] at which the vertex (maximum point) occurs can be calculated using the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
- Substitute the coefficients into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = 1.5 \][/tex]
3. Calculate 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]
[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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