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Answer :
To find the maximum height of the projectile, we can use the given equation that describes its path:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a quadratic function, which forms a parabola when graphed. Since the coefficient of [tex]\( t^2 \)[/tex] is negative (-16), the parabola opens downward. The maximum height of the projectile corresponds to the vertex of this parabola.
To find the time at which the projectile reaches this maximum height, we use the formula for the vertex of a quadratic equation, which is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
Substituting these values into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
So, the projectile reaches its maximum height at [tex]\( t = 1.5 \)[/tex] seconds.
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]
First, calculate [tex]\( (1.5)^2 \)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
Now, substitute back into the equation:
[tex]\[ h(1.5) = -16 \times 2.25 + 48 \times 1.5 + 190 \][/tex]
Calculate each term:
1. [tex]\(-16 \times 2.25 = -36\)[/tex]
2. [tex]\(48 \times 1.5 = 72\)[/tex]
Then, add them:
[tex]\[ h(1.5) = -36 + 72 + 190 = 226 \][/tex]
Therefore, the maximum height of the projectile is 226 feet. The correct answer is:
226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
This equation is a quadratic function, which forms a parabola when graphed. Since the coefficient of [tex]\( t^2 \)[/tex] is negative (-16), the parabola opens downward. The maximum height of the projectile corresponds to the vertex of this parabola.
To find the time at which the projectile reaches this maximum height, we use the formula for the vertex of a quadratic equation, which is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 48 \)[/tex]
Substituting these values into the formula:
[tex]\[ t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
So, the projectile reaches its maximum height at [tex]\( t = 1.5 \)[/tex] seconds.
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]
First, calculate [tex]\( (1.5)^2 \)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
Now, substitute back into the equation:
[tex]\[ h(1.5) = -16 \times 2.25 + 48 \times 1.5 + 190 \][/tex]
Calculate each term:
1. [tex]\(-16 \times 2.25 = -36\)[/tex]
2. [tex]\(48 \times 1.5 = 72\)[/tex]
Then, add them:
[tex]\[ h(1.5) = -36 + 72 + 190 = 226 \][/tex]
Therefore, the maximum height of the projectile is 226 feet. The correct answer is:
226 feet.
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