The image contains three math problems. I'll solve each one and provide the answers with explanations.

Problem 1:

$2(2x + 3) = 4x + 8$ We are asked to find the value of $x$ that satisfies this equation.

Solution:

1. Distribute the $2$ on the left side: $4x + 6 = 4x + 8$
2. Subtract $4x$ from both sides: $6 = 8$ This is a contradiction, which means the equation has no solution.

Problem 2:

This problem involves analyzing a graph and solving based on its features. The graph represents a parabola.

Problem Statement: Determine the range of $x$ where the height $y$ of the projectile is at least 8 meters.

Solution:

• The graph shows a parabola with its vertex at $x = 3$ and $y = 10$ meters.
• The height of 8 meters corresponds to points where the parabola crosses $y = 8$.

By analyzing the graph:

• The parabola crosses $y = 8$ at approximately $x = 2$ and $x = 4$.
• Thus, the range of $x$ where the height is at least 8 meters is $2 \leq x \leq 4$.

Problem 3:

Problem Statement: Two poles, A and B, are located 60 meters apart. The height of pole A is 20 meters, and the height of pole B is 40 meters. A rope is tied from the top of pole A to the top of pole B. What is the height of the rope when it is at a point directly above the point on the ground that is 30 meters from both poles?

Solution: This is a classic application of similar triangles.

Let's denote the height of the rope above the ground at the midpoint between the two poles as $h$. The midpoint is 30 meters from each pole.

1. Since the point is equidistant from both poles: $\text{Using the similarity of triangles: } \frac{h}{30} = \frac{40 - 20}{60}$
2. Simplifying: $\frac{h}{30} = \frac{20}{60} = \frac{1}{3}$ So, $h = 10$ meters.

Thus, the height of the rope at that point is 10 meters.

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If you need further details on any of these solutions or have any questions, feel free to ask!

Relative Questions:

1. How do you determine when a quadratic equation has no solution?
2. What are the properties of a parabola that help in solving graph-based questions?
3. How do you apply the concept of similar triangles in real-life problems?
4. What is the significance of the vertex in a parabolic graph?
5. How do you solve linear equations with variables on both sides?
6. What is the range and how is it determined in the context of a graph?
7. How can contradictions in algebraic equations be identified?
8. How can you find the point where a rope is at its lowest height between two poles?

Tip:

Always check for contradictions when solving equations; if both sides simplify to a false statement, the equation has no solution.