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Answer :
To find the solutions for the rocket's height problem, we will go through each part step by step:
1. Finding the Maximum Height:
The height of the rocket is given by the quadratic equation:
[tex]\[ h = -16t^2 + 112t + 704 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 112 \)[/tex], and [tex]\( c = 704 \)[/tex].
To find the maximum height, we use the formula to find the vertex of a quadratic equation, which gives us the time [tex]\( t \)[/tex] at the maximum height:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values, we get:
[tex]\[ t = -\frac{112}{2 \times -16} = 3.5 \][/tex] seconds.
Now, substitute [tex]\( t = 3.5 \)[/tex] back into the height equation to find the maximum height:
[tex]\[ h = -16(3.5)^2 + 112(3.5) + 704 \][/tex]
[tex]\[ h = 900 \][/tex] feet.
So, the maximum height of the rocket is 900 feet, and it is reached at 3.5 seconds.
2. Finding When the Rocket Reaches the Ground:
The rocket reaches the ground when the height [tex]\( h = 0 \)[/tex]. So, we set the equation to zero and solve for [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 112t + 704 = 0 \][/tex]
This is a quadratic equation, and we use the quadratic formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\( a = -16 \)[/tex], [tex]\( b = 112 \)[/tex], and [tex]\( c = 704 \)[/tex] into the formula:
- Calculate the discriminant:
[tex]\[ b^2 - 4ac = 112^2 - 4 \times (-16) \times 704 \][/tex]
- Find the roots using the quadratic formula, and you will find two values for [tex]\( t \)[/tex]. From these, choose the positive root, which represents the time after launch when the rocket hits the ground.
After performing the calculation, the rocket reaches the ground at 11 seconds.
Therefore, after being fired, the rocket reaches its maximum height of 900 feet at 3.5 seconds and hits the ground at 11 seconds.
1. Finding the Maximum Height:
The height of the rocket is given by the quadratic equation:
[tex]\[ h = -16t^2 + 112t + 704 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 112 \)[/tex], and [tex]\( c = 704 \)[/tex].
To find the maximum height, we use the formula to find the vertex of a quadratic equation, which gives us the time [tex]\( t \)[/tex] at the maximum height:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values, we get:
[tex]\[ t = -\frac{112}{2 \times -16} = 3.5 \][/tex] seconds.
Now, substitute [tex]\( t = 3.5 \)[/tex] back into the height equation to find the maximum height:
[tex]\[ h = -16(3.5)^2 + 112(3.5) + 704 \][/tex]
[tex]\[ h = 900 \][/tex] feet.
So, the maximum height of the rocket is 900 feet, and it is reached at 3.5 seconds.
2. Finding When the Rocket Reaches the Ground:
The rocket reaches the ground when the height [tex]\( h = 0 \)[/tex]. So, we set the equation to zero and solve for [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 112t + 704 = 0 \][/tex]
This is a quadratic equation, and we use the quadratic formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\( a = -16 \)[/tex], [tex]\( b = 112 \)[/tex], and [tex]\( c = 704 \)[/tex] into the formula:
- Calculate the discriminant:
[tex]\[ b^2 - 4ac = 112^2 - 4 \times (-16) \times 704 \][/tex]
- Find the roots using the quadratic formula, and you will find two values for [tex]\( t \)[/tex]. From these, choose the positive root, which represents the time after launch when the rocket hits the ground.
After performing the calculation, the rocket reaches the ground at 11 seconds.
Therefore, after being fired, the rocket reaches its maximum height of 900 feet at 3.5 seconds and hits the ground at 11 seconds.
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