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Answer :
To determine when the toy rocket is higher than 25 meters above the ground, we can follow these steps:
1. Understand the Function: The height of the rocket is described by the equation [tex]\( h = -4.9t^2 + 19.6t + 11 \)[/tex], where [tex]\( h \)[/tex] is the height in meters and [tex]\( t \)[/tex] is the time in seconds.
2. Set Up the Inequality: We want to find when the height [tex]\( h \)[/tex] is greater than 25 meters. Therefore, we set up the inequality:
[tex]\[
-4.9t^2 + 19.6t + 11 > 25
\][/tex]
3. Simplify the Inequality: Subtract 25 from both sides to simplify:
[tex]\[
-4.9t^2 + 19.6t + 11 - 25 > 0
\][/tex]
Simplifying gives:
[tex]\[
-4.9t^2 + 19.6t - 14 > 0
\][/tex]
4. Solve the Quadratic Equation: We need to find the values of [tex]\( t \)[/tex] by solving the equation [tex]\( -4.9t^2 + 19.6t - 14 = 0 \)[/tex]. Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -4.9 \)[/tex], [tex]\( b = 19.6 \)[/tex], and [tex]\( c = -14 \)[/tex], we can find the roots for [tex]\( t \)[/tex].
5. Determine the Interval: The rocket is above 25 meters between the two times, which are the roots of the equation. These roots represent the times the rocket passes 25 meters on the way up and on the way down.
6. Result: Based on the determination, the rocket is higher than 25 meters between [tex]\( t = 0.9 \)[/tex] seconds and [tex]\( t = 3.1 \)[/tex] seconds.
Therefore, the rocket is above 25 meters from approximately 0.9 seconds after launch to 3.1 seconds after launch.
1. Understand the Function: The height of the rocket is described by the equation [tex]\( h = -4.9t^2 + 19.6t + 11 \)[/tex], where [tex]\( h \)[/tex] is the height in meters and [tex]\( t \)[/tex] is the time in seconds.
2. Set Up the Inequality: We want to find when the height [tex]\( h \)[/tex] is greater than 25 meters. Therefore, we set up the inequality:
[tex]\[
-4.9t^2 + 19.6t + 11 > 25
\][/tex]
3. Simplify the Inequality: Subtract 25 from both sides to simplify:
[tex]\[
-4.9t^2 + 19.6t + 11 - 25 > 0
\][/tex]
Simplifying gives:
[tex]\[
-4.9t^2 + 19.6t - 14 > 0
\][/tex]
4. Solve the Quadratic Equation: We need to find the values of [tex]\( t \)[/tex] by solving the equation [tex]\( -4.9t^2 + 19.6t - 14 = 0 \)[/tex]. Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -4.9 \)[/tex], [tex]\( b = 19.6 \)[/tex], and [tex]\( c = -14 \)[/tex], we can find the roots for [tex]\( t \)[/tex].
5. Determine the Interval: The rocket is above 25 meters between the two times, which are the roots of the equation. These roots represent the times the rocket passes 25 meters on the way up and on the way down.
6. Result: Based on the determination, the rocket is higher than 25 meters between [tex]\( t = 0.9 \)[/tex] seconds and [tex]\( t = 3.1 \)[/tex] seconds.
Therefore, the rocket is above 25 meters from approximately 0.9 seconds after launch to 3.1 seconds after launch.
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