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Answer :
To find the maximum height of the projectile, we can use the equation that describes its motion:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a quadratic function, which forms the shape of a parabola. The maximum height is found at the highest point of this parabola, known as its vertex.
### Steps to Find the Maximum Height
1. Identify the coefficients:
The general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. In this problem:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
2. Calculate the time at which the maximum height occurs:
The time [tex]\( t \)[/tex] at which the maximum height occurs is given by the formula for the vertex of a parabola:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
t = -\frac{48}{2 \times (-16)}
\][/tex]
[tex]\[
t = -\frac{48}{-32} = 1.5 \text{ seconds}
\][/tex]
4. Find the maximum height by substituting [tex]\( t \)[/tex] back into the height equation:
[tex]\[
h(t) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
5. Calculate the height:
[tex]\[
h(1.5) = -16 \times 2.25 + 72 + 190
\][/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 quadratic function, which forms the shape of a parabola. The maximum height is found at the highest point of this parabola, known as its vertex.
### Steps to Find the Maximum Height
1. Identify the coefficients:
The general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. In this problem:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 190 \)[/tex]
2. Calculate the time at which the maximum height occurs:
The time [tex]\( t \)[/tex] at which the maximum height occurs is given by the formula for the vertex of a parabola:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
t = -\frac{48}{2 \times (-16)}
\][/tex]
[tex]\[
t = -\frac{48}{-32} = 1.5 \text{ seconds}
\][/tex]
4. Find the maximum height by substituting [tex]\( t \)[/tex] back into the height equation:
[tex]\[
h(t) = -16(1.5)^2 + 48(1.5) + 190
\][/tex]
5. Calculate the height:
[tex]\[
h(1.5) = -16 \times 2.25 + 72 + 190
\][/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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