Final answer:
To find the length of the sides of the triangle, set up an equation based on the information provided. By solving this equation, we find the middle side to be approximately 924.66 miles, the shortest side to be 846.66 miles, and the longest side to be 1302.66 miles.
Step-by-step explanation:
To find the lengths of the sides of the triangle, you first need to establish variables for each side. Let's assume the middle side of the triangle is x. Therefore, the shortest side will be x - 78 (since it's 78 mi less than the middle side), and the longest side will be x + 378 (as it's 378 mi more than the middle side).
Since the triangle's perimeter is the sum of the lengths of all sides, you can construct the equation x + (x - 78) + (x + 378) = 3074. Simplifying this equation gives 3x + 300 = 3074. Subtract 300 from both sides to isolate the terms with x, resulting in 3x = 2774. To find the value of x, divide both sides by 3, yielding x = 924.66 (rounded to two decimal places). This is the middle side's length.
Implementing these values, the shortest side will be 924.66 - 78 = 846.66 miles, and the longest side will be 924.66 + 378 = 1302.66 miles. So, the lengths of the three sides are approximately 924.66 mi, 846.66 mi, and 1302.66 mi.
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