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Answer :
To find the maximum height of the projectile, you need to determine the vertex of the parabola represented by the equation [tex]\( h(t) = -16t^2 + 48t + 190 \)[/tex]. This equation models the height [tex]\( h \)[/tex] of the projectile over time [tex]\( t \)[/tex], where:
- [tex]\( h(t) \)[/tex] is the height in feet,
- [tex]\( t \)[/tex] is the time in seconds,
- [tex]\(-16t^2\)[/tex] represents the effect of gravity,
- [tex]\( 48t \)[/tex] represents the initial upward velocity,
- [tex]\( 190 \)[/tex] is the initial height of the building in feet.
In a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the maximum or minimum value of the function (the vertex) occurs at [tex]\( t = -\frac{b}{2a} \)[/tex].
### Step-by-step Solution:
1. Identify the coefficients:
- Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Calculate the time at which the maximum height occurs:
- Use the formula for the vertex:
[tex]\[
t = -\frac{b}{2a} = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \text{ seconds}
\][/tex]
3. Substitute this time back into the height equation to find the maximum height:
- Calculate [tex]\( h(1.5) \)[/tex]:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
[tex]\[
= -16(2.25) + 72 + 190
\][/tex]
[tex]\[
= -36 + 72 + 190
\][/tex]
[tex]\[
= 226 \text{ feet}
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
- [tex]\( h(t) \)[/tex] is the height in feet,
- [tex]\( t \)[/tex] is the time in seconds,
- [tex]\(-16t^2\)[/tex] represents the effect of gravity,
- [tex]\( 48t \)[/tex] represents the initial upward velocity,
- [tex]\( 190 \)[/tex] is the initial height of the building in feet.
In a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the maximum or minimum value of the function (the vertex) occurs at [tex]\( t = -\frac{b}{2a} \)[/tex].
### Step-by-step Solution:
1. Identify the coefficients:
- Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
2. Calculate the time at which the maximum height occurs:
- Use the formula for the vertex:
[tex]\[
t = -\frac{b}{2a} = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \text{ seconds}
\][/tex]
3. Substitute this time back into the height equation to find the maximum height:
- Calculate [tex]\( h(1.5) \)[/tex]:
[tex]\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
[tex]\[
= -16(2.25) + 72 + 190
\][/tex]
[tex]\[
= -36 + 72 + 190
\][/tex]
[tex]\[
= 226 \text{ feet}
\][/tex]
Therefore, the maximum height of the projectile is 226 feet.
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