The captain of a boat is planning to travel to three islands in a triangular pattern where one side of the side island is 75 miles and the other side is 32 miles. What is the possible range for the number of miles round trip the boat will travel?
Answer: between 150 and 214 miles
Please explain how did they get the answer. Thanks
Not quite sure what
"one side of the side island"
means. However, if you mean that one side of the triangle is 75 and the other is 32, then we know that the third side x must obey
75-32 < x < 75+32
43 < x < 107
Now use that to see the bounds of the perimeter (the sum of all 3 sides)
hi sally heh
To find the range of miles for the round trip, we need to consider the different possible routes the boat can take between the islands.
Let's first visualize the triangle formed by the three islands. We have two sides given: one side is 75 miles, and the other side is 32 miles. To determine the range of the round trip, we need to consider the total distance traveled in both directions.
Now, let's examine the possible paths the boat can take.
Case 1: The boat travels directly between the islands.
In this case, the boat would travel from Island A to Island B (75 miles), Island B to Island C (32 miles), and then back from Island C to Island A (75 miles). So the total round trip distance would be 75 + 32 + 75 = 182 miles.
Case 2: The boat takes a detour.
In this case, the boat would travel from Island A to Island C (across the longer side - 75 miles), then from Island C to Island B (across the shorter side - 32 miles), and finally back from Island B to Island A (across the remaining side - 75 miles). So the total round trip distance would be 75 + 32 + 75 = 182 miles.
The maximum round trip distance is 182 miles (found in both cases).
To find the minimum round trip distance, we need to determine the shortest possible path between the islands. In this case, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
So for the shortest path, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's consider Island A as the starting point and look at the combinations:
1. Combination 1: Island A to Island B (75 miles), which means the sum of the two sides is 75 + 32 = 107, which is greater than the third side that is 75 miles. This combination is valid.
2. Combination 2: Island A to Island C (75 miles), which means the sum of the two sides is 75 + 75 = 150, but this is not greater than the third side, which is also 75 miles. This combination is not valid.
Thus, the minimum round trip distance is the sum of the three side lengths of the triangle, which is 75 + 32 + 75 = 182 miles.
Therefore, the possible range for the number of miles round trip the boat will travel is between 150 and 214 miles (minimum of 182 miles and maximum of 182 miles).