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Answer :
To find the maximum height of a projectile, we need to examine its height equation, which is given as:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
### Step-by-Step Solution:
1. Identify the Quadratic Equation:
The equation for the projectile's height is 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]
2. Determine the Time at Maximum Height:
The time at which the maximum height is reached for a projectile's path modeled by a quadratic equation is found using the formula for the vertex of a parabola:
[tex]\[
t_{\text{max}} = -\frac{b}{2a}
\][/tex]
Substituting the values we have:
[tex]\[
t_{\text{max}} = -\frac{48}{2 \times (-16)} = \frac{48}{32} = 1.5
\][/tex]
3. Calculate the Maximum Height:
Substitute [tex]\( t_{\text{max}} = 1.5 \)[/tex] into the height equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
Calculating each term step-by-step:
- [tex]\(-16 \times (1.5)^2 = -16 \times 2.25 = -36\)[/tex]
- [tex]\(48 \times 1.5 = 72\)[/tex]
- Adding these to the initial height: [tex]\(-36 + 72 + 190 = 226\)[/tex]
Therefore, the maximum height of the projectile is [tex]\( 226 \)[/tex] feet.
In conclusion, the projectile reaches a maximum height of 226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
### Step-by-Step Solution:
1. Identify the Quadratic Equation:
The equation for the projectile's height is 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]
2. Determine the Time at Maximum Height:
The time at which the maximum height is reached for a projectile's path modeled by a quadratic equation is found using the formula for the vertex of a parabola:
[tex]\[
t_{\text{max}} = -\frac{b}{2a}
\][/tex]
Substituting the values we have:
[tex]\[
t_{\text{max}} = -\frac{48}{2 \times (-16)} = \frac{48}{32} = 1.5
\][/tex]
3. Calculate the Maximum Height:
Substitute [tex]\( t_{\text{max}} = 1.5 \)[/tex] into the height equation to find the maximum height:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
Calculating each term step-by-step:
- [tex]\(-16 \times (1.5)^2 = -16 \times 2.25 = -36\)[/tex]
- [tex]\(48 \times 1.5 = 72\)[/tex]
- Adding these to the initial height: [tex]\(-36 + 72 + 190 = 226\)[/tex]
Therefore, the maximum height of the projectile is [tex]\( 226 \)[/tex] feet.
In conclusion, the projectile reaches a maximum height of 226 feet.
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