Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Triangle Find the area of the Bermuda Triangle if the sides of the triangle have approximate lengths of 850 miles, 925 miles, and 1300 miles.

Step-by-Step Solution

Answer

The area is approximately 375,469 square miles.

Step 1: Verify the possibility of the triangle

To find the area of a triangle with given sides, first ensure that a triangle with these side lengths can exist. According to the triangle inequality theorem, the sum of the lengths of any two sides must be greater than the length of the third side. Check: - 850 + 925 > 1300 - 850 + 1300 > 925 - 925 + 1300 > 850 If all these conditions are satisfied, the triangle can exist.

Step 2: Calculate the semi-perimeter

Use the formula for the semi-perimeter, which is half the sum of the lengths of the sides of the triangle. Let's denote the sides as a = 850 miles, b = 925 miles, and c = 1300 miles.\[ s = \frac{a + b + c}{2} = \frac{850 + 925 + 1300}{2} \]Calculate this to find \( s \).

Step 3: Apply Heron's Formula

Heron's formula allows us to calculate the area of a triangle when the lengths of all its sides are known. The formula is:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the values from Step 2 and solve for the area.

Step 4: Simplify and calculate the area

Simplify the expression by calculating each term inside the square root, and then find the value of the area \( A \).Ensure you carefully compute each step:- Calculate \( s-a \), \( s-b \), and \( s-c \)- Multiply these values by \( s \)- Take the square root of the product to find \( A \)

Key Concepts

Triangle Inequality Theorem

When trying to determine whether a triangle can be formed from three given lengths, it's crucial to use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This condition ensures that the sides can form a closed shape. Here is how you can apply this concept:

• Check that the sum of two smallest sides is greater than the longest side.
• Ensure that each pair of side sums is greater than the remaining side.
• If all these conditions are met, a triangle can exist with those side lengths.

For example, with sides 850 miles, 925 miles, and 1300 miles, confirm: - 850 + 925 > 1300, - 850 + 1300 > 925, - 925 + 1300 > 850. Each sum being greater than the third side confirms the possibility of creating a triangle.

Semi-perimeter

The semi-perimeter of a triangle is a very useful concept when applying Heron's formula. It is defined as half the sum of the triangle's side lengths. Understanding this concept is crucial because it simplifies the process of finding the area.Here's how to calculate it step by step:

• First, add up all the side lengths of the triangle.
• Then, divide this sum by two to find the semi-perimeter.

For a triangle with sides 850 miles, 925 miles, and 1300 miles, the semi-perimeter can be calculated as:\[ s = \frac{850 + 925 + 1300}{2} = 1537.5 \text{ miles}\]With practice, finding the semi-perimeter becomes an easy and quick task, laying the groundwork for repetitive problem solving using Heron's formula.

Area of a Triangle

Finding the area of a triangle when all three sides are known is made efficient by Heron's formula. After confirming the triangle can exist and finding the semi-perimeter, you use this powerful formula for the area:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]In this formula:

• \( s \) is the semi-perimeter;
• \( a, b, c \) are the side lengths of the triangle.

For the given triangle:- Calculate each difference: - \( s-a = 1537.5 - 850 = 687.5 \) - \( s-b = 1537.5 - 925 = 612.5 \) - \( s-c = 1537.5 - 1300 = 237.5 \)- Multiply these results by the semi-perimeter \( s \).- Finally, take the square root of the computed product to find the area \( A \).This calculation provides the area, giving a precise understanding of the space contained within these triangle sides.