A ladder leans against a wall. The angle of elevation of the top of the ladder from the foot of the wall is 60°. The foot of the ladder is pulled 3.5 meters away from the wall, and now the angle of elevation is 30°. Find the length of the ladder.

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We have a ladder problem involving two positions. Initially, the ladder makes a 60-degree angle with the ground. When the foot of the ladder is pulled 3.5 meters away from the wall, the angle becomes 30 degrees. We need to find the length of the ladder, which remains constant in both positions. Let's analyze the first position where the angle of elevation is 60 degrees. In this right triangle, we can use trigonometric ratios. The sine of 60 degrees equals the height h divided by the ladder length L. Since sine of 60 degrees is square root of 3 over 2, we get h equals L times square root of 3 over 2. Similarly, cosine of 60 degrees equals the base distance d1 divided by L, giving us d1 equals L over 2. Now let's analyze the second position where the angle of elevation is 30 degrees. The ladder has been pulled 3.5 meters away from the wall. Using trigonometry again, sine of 30 degrees equals h over L. Since sine of 30 degrees is one half, we get h equals L over 2. Cosine of 30 degrees equals the new base distance d2 over L. Since cosine of 30 degrees is square root of 3 over 2, we get d2 equals L times square root of 3 over 2. The key relationship is that d2 equals d1 plus 3.5 meters. Now let's connect the two positions by analyzing the horizontal distances. From the first position, we know that cosine of 60 degrees equals d1 over L, which gives us d1 equals L over 2. From the second position, cosine of 30 degrees equals d2 over L, which gives us d2 equals L times square root of 3 over 2. The key constraint is that d2 minus d1 equals 3.5 meters, representing the distance the ladder was pulled away from the wall. Now let's solve the system of equations. We start with the constraint that d2 minus d1 equals 3.5 meters. Substituting our expressions, we get L times square root of 3 over 2 minus L over 2 equals 3.5. Factoring out L, we have L times the quantity square root of 3 minus 1, all over 2, equals 3.5. Multiplying both sides by 2, we get L times the quantity square root of 3 minus 1 equals 7. Therefore, L equals 7 divided by the quantity square root of 3 minus 1. To rationalize the denominator, we multiply both numerator and denominator by square root of 3 plus 1, giving us L equals 7 times the quantity square root of 3 plus 1, all over 2.