A Boat Initially Traveled Southwest For 25 Miles Before Stopping. When It Stopped, The Boat Was 18 Miles From Its Starting Point.By How Many Degrees Did The Direction Of The Boat Change When It Made Its First Turn? Round To The Nearest Degree.A. 30
Introduction
In this article, we will delve into a mathematical problem involving a boat's change in direction. The problem is as follows: a boat initially traveled southwest for 25 miles before stopping. When it stopped, the boat was 18 miles from its starting point. We will use trigonometry to determine by how many degrees the direction of the boat changed when it made its first turn.
Understanding the Problem
To solve this problem, we need to understand the concept of trigonometry, particularly the relationship between angles and sides of a right-angled triangle. We will use the Pythagorean theorem to find the length of the third side of the triangle and then use trigonometric ratios to determine the angle of the boat's direction.
Visualizing the Problem
Let's visualize the problem by drawing a diagram. We can draw a right-angled triangle with the boat's initial direction as the hypotenuse (the side opposite the right angle). The length of the hypotenuse is 25 miles, and the length of one of the other sides (the side opposite the angle we want to find) is 18 miles.
+---------------+
| |
| 25 miles |
| (hypotenuse)|
| |
+---------------+
| |
| 18 miles |
| (side) |
| |
+---------------+
Using the Pythagorean Theorem
We can use the Pythagorean theorem to find the length of the third side of the triangle. The Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In this case, we know the length of the hypotenuse (25 miles) and one of the other sides (18 miles). We can use the theorem to find the length of the third side.
a^2 + b^2 = c^2
18^2 + b^2 = 25^2
324 + b^2 = 625
b^2 = 625 - 324
b^2 = 301
b = sqrt(301)
b ≈ 17.34 miles
Finding the Angle
Now that we have the length of the third side, we can use trigonometric ratios to find the angle of the boat's direction. We can use the sine, cosine, or tangent ratio to find the angle. In this case, we will use the tangent ratio.
tan(angle) = opposite side / adjacent side
tan(angle) = 18 miles / 17.34 miles
angle = arctan(18/17.34)
angle ≈ 48.59 degrees
Finding the Change in Direction
The problem asks us to find the change in direction of the boat when it made its first turn. We can find the change in direction by subtracting the initial angle from the final angle.
change in direction = final angle - initial angle
change in direction = 48.59 degrees - 135 degrees
change in direction ≈ -86.41 degrees
However, since we are asked to round to the nearest degree, we can round the change in direction to -86 degrees.
Conclusion
In this article, we used trigonometry to solve a problem involving a boat's change in direction. We used the Pythagorean theorem to find the length of the third side of a right-angled triangle and then used trigonometric ratios to find the angle of the boat's direction. We found that the change in direction of the boat when it made its first turn was approximately -86 degrees.
Final Answer
Introduction
In our previous article, we explored a mathematical problem involving a boat's change in direction. We used trigonometry to determine by how many degrees the direction of the boat changed when it made its first turn. In this article, we will answer some frequently asked questions related to the problem.
Q: What is the initial direction of the boat?
A: The initial direction of the boat is southwest.
Q: How far did the boat travel before stopping?
A: The boat traveled 25 miles before stopping.
Q: How far was the boat from its starting point when it stopped?
A: The boat was 18 miles from its starting point when it stopped.
Q: What is the relationship between the angle and the sides of a right-angled triangle?
A: The angle and the sides of a right-angled triangle are related by trigonometric ratios. The sine, cosine, and tangent ratios can be used to find the angle of a right-angled triangle.
Q: How did you find the length of the third side of the triangle?
A: We used the Pythagorean theorem to find the length of the third side of the triangle. The Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Q: What is the tangent ratio and how did you use it to find the angle?
A: The tangent ratio is defined as the ratio of the opposite side to the adjacent side. We used the tangent ratio to find the angle of the boat's direction. We calculated the tangent of the angle using the lengths of the opposite and adjacent sides.
Q: How did you find the change in direction of the boat?
A: We found the change in direction of the boat by subtracting the initial angle from the final angle. However, since we are asked to round to the nearest degree, we rounded the change in direction to -86 degrees.
Q: What is the significance of the change in direction of the boat?
A: The change in direction of the boat is significant because it indicates the amount of turn the boat made when it changed direction. In this case, the boat changed direction by approximately 86 degrees.
Q: Can you provide a diagram to illustrate the problem?
A: Yes, we can provide a diagram to illustrate the problem. Here is a diagram of the right-angled triangle:
+---------------+
| |
| 25 miles |
| (hypotenuse)|
| |
+---------------+
| |
| 18 miles |
| (side) |
| |
+---------------+
Conclusion
In this article, we answered some frequently asked questions related to the problem of a boat's change in direction. We provided explanations and examples to help clarify the concepts and calculations involved in solving the problem.
Final Answer
The final answer is: -86