Raul is laying a cement concrete walkway in his backyard. He decides to put a turn of 30° from the horizontal in the walkway to go around a tree and flowerbed. Once he gets past the tree, he wants to continue the walkway in the same direction as before. At what angle should he put the second turn so that the walkway will continue in the same direction as before he came to the flowerbed?
Not sure how he can turn from the horizontal, unless he goes uphill, but whatever the direction, just turning 30° in the opposite manner will put him on a course parallel to his original one.
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Well, Raul certainly has a tricky situation on his hands! If he wants the walkway to continue in the same direction as before he came to the flowerbed, he needs to make sure the second turn aligns with the original direction. To figure out the angle, we can use the fact that the sum of the angles in a triangle is 180°.
So, let's break it down. Raul already made a 30° turn from the horizontal at the first turn. To continue in the same direction, he needs to make another turn that balances out the first turn. Therefore, the second turn should be -30° (the negative sign indicates that it's in the opposite direction) to cancel out the first turn.
By making a -30° turn at the second turn, Raul should be able to continue the walkway in the same direction as before he came to the flowerbed. Just make sure to remind him to keep calm and not get too twisted up in his own walkway!
To determine the angle at which Raul should put the second turn so that the walkway continues in the same direction as before coming to the flowerbed, we need to consider the total change in direction caused by the turns.
Given that Raul turned 30° from the horizontal to go around the tree, we can visualize this as a rotation of 30° counterclockwise.
To continue the walkway in the same direction as before, the second turn should compensate for the initial 30° turn. Since the first turn was 30° counterclockwise, the second turn should be 30° clockwise to cancel out the previous rotation.
Therefore, Raul should put the second turn at an angle of 30° clockwise to ensure that the walkway continues in the same direction as before coming to the flowerbed.
To answer this question, we need to understand the geometry involved. Let's break it down step by step:
1. Draw a diagram: Visualizing the situation often helps in solving problems like these. Start by drawing a straight line to represent the initial direction of the walkway.
2. Add the first turn: Place the line representing the tree and flowerbed at an angle of 30° to the horizontal. This represents the first turn that Raul wants to make around the tree.
3. Determine the initial and final directions: Now, we need to find the angle at which the walkway should continue after passing the tree, so it remains in the same direction as before reaching the flowerbed.
4. Use the idea of supplementary angles: Remember that supplementary angles add up to 180°. Since Raul wants the walkway to continue in the same direction, the final angle (after transitioning from the first turn) and the initial angle (before the first turn) should be supplementary.
5. Calculation: Let's assume the initial direction of the walkway is represented by an angle of x°. According to the supplementary rule, the final angle should be (180° - x°) to maintain the same direction.
So, in order to have the walkway continue in the same direction as before reaching the flowerbed, Raul should put the second turn at an angle of (180° - x°) to the horizontal.