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Answer :
To find the maximum height of a projectile, we need to analyze the quadratic equation provided, which models the path of the projectile:
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
In this equation:
- [tex]\( h(t) \)[/tex] represents the height of the projectile at any time [tex]\( t \)[/tex].
- The coefficients are: [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
This is a quadratic equation, and its graph is a parabola. Since the coefficient of [tex]\( t^2 \)[/tex] (which is [tex]\( a = -16 \)[/tex]) is negative, the parabola opens downwards. This means the vertex of the parabola will give us the maximum height.
The formula to find the time [tex]\( t \)[/tex] at which the maximum height occurs is given by the vertex formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula:
[tex]\[ t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
Now that we have the time [tex]\( t = 1.5 \)[/tex] seconds when the projectile reaches its maximum height, we substitute this back into the equation for [tex]\( h(t) \)[/tex] to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculate step-by-step:
1. [tex]\((1.5)^2 = 2.25\)[/tex]
2. [tex]\(-16 \times 2.25 = -36\)[/tex]
3. [tex]\(48 \times 1.5 = 72\)[/tex]
4. Combine these results: [tex]\( h(1.5) = -36 + 72 + 190\)[/tex]
5. Simplify: [tex]\( h(1.5) = 226\)[/tex]
Thus, the maximum height of the projectile is 226 feet.
[tex]\[ h(t) = -16t^2 + 48t + 190 \][/tex]
In this equation:
- [tex]\( h(t) \)[/tex] represents the height of the projectile at any time [tex]\( t \)[/tex].
- The coefficients are: [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 190 \)[/tex].
This is a quadratic equation, and its graph is a parabola. Since the coefficient of [tex]\( t^2 \)[/tex] (which is [tex]\( a = -16 \)[/tex]) is negative, the parabola opens downwards. This means the vertex of the parabola will give us the maximum height.
The formula to find the time [tex]\( t \)[/tex] at which the maximum height occurs is given by the vertex formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula:
[tex]\[ t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
Now that we have the time [tex]\( t = 1.5 \)[/tex] seconds when the projectile reaches its maximum height, we substitute this back into the equation for [tex]\( h(t) \)[/tex] to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 190 \][/tex]
Calculate step-by-step:
1. [tex]\((1.5)^2 = 2.25\)[/tex]
2. [tex]\(-16 \times 2.25 = -36\)[/tex]
3. [tex]\(48 \times 1.5 = 72\)[/tex]
4. Combine these results: [tex]\( h(1.5) = -36 + 72 + 190\)[/tex]
5. Simplify: [tex]\( h(1.5) = 226\)[/tex]
Thus, the maximum height of the projectile is 226 feet.
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