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Answer :
To find the height of the rocket above the ground when the observer sees its tip at an angle of [tex]\( q \)[/tex] from the horizontal, we can use some basic trigonometry.
Here's the step-by-step solution:
1. Understand the Scenario:
- An observer is standing [tex]\( x \)[/tex] meters from the base of a rocket launch pad.
- The rocket takes off vertically and, at a certain moment, the observer sees the tip of the rocket at an angle [tex]\( q \)[/tex] from the horizontal.
2. Visualize the Situation:
- This forms a right triangle where:
- The horizontal distance (the base of the triangle) is [tex]\( x \)[/tex].
- The vertical distance (the height of the rocket above the ground) is what we are looking for.
- The angle [tex]\( q \)[/tex] is the angle of elevation from the observer to the tip of the rocket.
3. Use Trigonometry:
- In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
- For this triangle, the opposite side is the height of the rocket, and the adjacent side is [tex]\( x \)[/tex].
4. Set up the Tangent Function:
[tex]\[
\tan(q) = \frac{\text{height}}{x}
\][/tex]
5. Solve for the Height:
- Rearrange the equation to solve for the height:
[tex]\[
\text{height} = x \times \tan(q)
\][/tex]
6. Select the Correct Answer:
- Comparing this result with the options given:
- Option C, [tex]\( x \tan q \)[/tex], matches our calculated expression.
Therefore, the height of the rocket above the ground at that instant is [tex]\( x \tan q \)[/tex].
Here's the step-by-step solution:
1. Understand the Scenario:
- An observer is standing [tex]\( x \)[/tex] meters from the base of a rocket launch pad.
- The rocket takes off vertically and, at a certain moment, the observer sees the tip of the rocket at an angle [tex]\( q \)[/tex] from the horizontal.
2. Visualize the Situation:
- This forms a right triangle where:
- The horizontal distance (the base of the triangle) is [tex]\( x \)[/tex].
- The vertical distance (the height of the rocket above the ground) is what we are looking for.
- The angle [tex]\( q \)[/tex] is the angle of elevation from the observer to the tip of the rocket.
3. Use Trigonometry:
- In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
- For this triangle, the opposite side is the height of the rocket, and the adjacent side is [tex]\( x \)[/tex].
4. Set up the Tangent Function:
[tex]\[
\tan(q) = \frac{\text{height}}{x}
\][/tex]
5. Solve for the Height:
- Rearrange the equation to solve for the height:
[tex]\[
\text{height} = x \times \tan(q)
\][/tex]
6. Select the Correct Answer:
- Comparing this result with the options given:
- Option C, [tex]\( x \tan q \)[/tex], matches our calculated expression.
Therefore, the height of the rocket above the ground at that instant is [tex]\( x \tan q \)[/tex].
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