solve---**Question 6** **Question Stem:** Find the exact value of x. **Chart/Diagram Description:** * **Type:** Geometric figure, specifically a composite of two right-angled triangles. * **Main Elements:** * **Vertical Line Segment:** Represents the height of the triangle. It is labeled "x m". * **Horizontal Base Line:** Forms the base of the triangles. * **Right Angle:** Indicated by a square symbol at the junction of the vertical line and the horizontal base, signifying a 90° angle. * **Hypotenuses:** Two diagonal lines connect the top of the vertical line to two distinct points on the horizontal base. * **Angles:** * The angle of elevation from the point closer to the vertical line is 45°. * The angle of elevation from the point further away from the vertical line is 30°. * **Lengths/Distances:** * The height is denoted by "x m". * The horizontal distance between the two points on the base where the 45° and 30° angles are measured is 100 m. This 100 m segment is located between the vertex corresponding to the 45° angle and the vertex corresponding to the 30° angle. * **Relative Position:** The two triangles share the vertical side "x m" as a common height. The 45° angle is associated with the smaller right-angled triangle, and the 30° angle is associated with the larger right-angled triangle. The 100 m segment is the difference in the base lengths of these two triangles.

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We need to find the exact value of x in this geometric problem. We have a composite triangle structure with a vertical height of x meters, two angles of elevation at 45 degrees and 30 degrees, and a horizontal distance of 100 meters between the two base points. Our goal is to use trigonometric relationships to solve for the unknown height x. Let's break down this composite figure into two separate right triangles that share the common height x. The first triangle has a 45-degree angle with base length d1, and the second triangle has a 30-degree angle with base length d2. The key relationship we need to establish is that d2 minus d1 equals 100 meters, which represents the horizontal distance between the two base points. Now let's apply trigonometric ratios to express the base lengths in terms of x. For the 45-degree triangle, tangent of 45 degrees equals x over d1. Since tangent of 45 degrees equals 1, we get d1 equals x. For the 30-degree triangle, tangent of 30 degrees equals x over d2. Since tangent of 30 degrees equals 1 over square root of 3, we get d2 equals x times square root of 3. Now let's combine our trigonometric relationships with the distance constraint to form a solvable equation. We start with our constraint that d2 minus d1 equals 100. Then we substitute our expressions: d1 equals x and d2 equals x times square root of 3. This gives us x times square root of 3 minus x equals 100. Finally, we factor out x to get x times the quantity square root of 3 minus 1 equals 100. Now let's solve the equation x times the quantity square root of 3 minus 1 equals 100. We divide both sides by square root of 3 minus 1 to get x equals 100 over square root of 3 minus 1. To rationalize the denominator, we multiply both numerator and denominator by square root of 3 plus 1. This gives us x equals 100 times square root of 3 plus 1, all over 2, which simplifies to our final answer: x equals 50 times the quantity square root of 3 plus 1.