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Answer :
To find out how long it will take for the toy rocket to hit the ground, we need to determine when its height, [tex]\( h \)[/tex], becomes zero. The height as a function of time, [tex]\( t \)[/tex], is given by the equation:
[tex]\[ h = -4.9t^2 + 14.0t + 2.1 \][/tex]
To find the time when the rocket hits the ground, set the height [tex]\( h \)[/tex] to zero and solve the equation:
[tex]\[ 0 = -4.9t^2 + 14.0t + 2.1 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where:
- [tex]\( a = -4.9 \)[/tex]
- [tex]\( b = 14.0 \)[/tex]
- [tex]\( c = 2.1 \)[/tex]
We can use the quadratic formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = (14.0)^2 - 4(-4.9)(2.1) \][/tex]
2. Compute the square root of the discriminant.
3. Apply the quadratic formula:
[tex]\[ t = \frac{{-14.0 \pm \sqrt{{(14.0)^2 - 4(-4.9)(2.1)}}}}{2(-4.9)} \][/tex]
After solving this, you'll find two values for [tex]\( t \)[/tex]. We discard the negative value because time cannot be negative in this context, and the remaining positive value represents the time when the rocket hits the ground.
The rocket will hit the ground after approximately 3.0 seconds.
[tex]\[ h = -4.9t^2 + 14.0t + 2.1 \][/tex]
To find the time when the rocket hits the ground, set the height [tex]\( h \)[/tex] to zero and solve the equation:
[tex]\[ 0 = -4.9t^2 + 14.0t + 2.1 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where:
- [tex]\( a = -4.9 \)[/tex]
- [tex]\( b = 14.0 \)[/tex]
- [tex]\( c = 2.1 \)[/tex]
We can use the quadratic formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = (14.0)^2 - 4(-4.9)(2.1) \][/tex]
2. Compute the square root of the discriminant.
3. Apply the quadratic formula:
[tex]\[ t = \frac{{-14.0 \pm \sqrt{{(14.0)^2 - 4(-4.9)(2.1)}}}}{2(-4.9)} \][/tex]
After solving this, you'll find two values for [tex]\( t \)[/tex]. We discard the negative value because time cannot be negative in this context, and the remaining positive value represents the time when the rocket hits the ground.
The rocket will hit the ground after approximately 3.0 seconds.
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