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Answer :
To find the maximum height of a projectile launched from a building, we use the given equation for the path of the projectile:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation represents a parabola, and the maximum height occurs at the vertex of the parabola.
To find the time [tex]\( t \)[/tex] at which the projectile reaches its maximum height, we use the formula for the vertex of a parabola:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
Substitute these values into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \text{ seconds} \][/tex]
Next, we substitute [tex]\( t = 1.5 \)[/tex] back into the original equation to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculate each term:
1. [tex]\( -16(1.5)^2 = -16 \times 2.25 = -36 \)[/tex]
2. [tex]\( 48 \times 1.5 = 72 \)[/tex]
3. [tex]\( 190 \)[/tex] (constant term)
Now, add these values together:
[tex]\[ h(1.5) = -36 + 72 + 190 \][/tex]
[tex]\[ h(1.5) = 226 \][/tex]
So, the maximum height of the projectile is 226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation represents a parabola, and the maximum height occurs at the vertex of the parabola.
To find the time [tex]\( t \)[/tex] at which the projectile reaches its maximum height, we use the formula for the vertex of a parabola:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
Substitute these values into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \text{ seconds} \][/tex]
Next, we substitute [tex]\( t = 1.5 \)[/tex] back into the original equation to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculate each term:
1. [tex]\( -16(1.5)^2 = -16 \times 2.25 = -36 \)[/tex]
2. [tex]\( 48 \times 1.5 = 72 \)[/tex]
3. [tex]\( 190 \)[/tex] (constant term)
Now, add these values together:
[tex]\[ h(1.5) = -36 + 72 + 190 \][/tex]
[tex]\[ h(1.5) = 226 \][/tex]
So, the maximum height of the projectile is 226 feet.
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