Problem 27 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=4.38 \mathrm{ft}, b=3.79 \mathrm{ft}, c=5.22 \mathrm{ft}\)

Step-by-Step Solution

Answer

The area of triangle ABC is approximately 8.14 square feet.

Step 1: Use Heron's Formula

To find the area of triangle using the side lengths, we use Heron's formula. Heron's formula states that the area of a triangle with sides of lengths \(a\), \(b\), and \(c\) is \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) is the semi-perimeter of the triangle.

Step 2: Calculate the Semi-Perimeter

First, calculate the semi-perimeter \(s\) using the formula: \(s = \frac{a + b + c}{2}\). Input the given values: \(a = 4.38\), \(b = 3.79\), \(c = 5.22\). Calculate \(s\): \[s = \frac{4.38 + 3.79 + 5.22}{2} = 6.695\].

Step 3: Substitute into Heron's Formula

Now substitute \(s\), \(a\), \(b\), and \(c\) into Heron's formula to find the area:\[A = \sqrt{6.695(6.695 - 4.38)(6.695 - 3.79)(6.695 - 5.22)}\].

Step 4: Simplify Each Term

Calculate each term:- \(s - a = 6.695 - 4.38 = 2.315\)- \(s - b = 6.695 - 3.79 = 2.905\)- \(s - c = 6.695 - 5.22 = 1.475\).

Step 5: Calculate the Area

Substitute these values back into the formula:\[A = \sqrt{6.695 \times 2.315 \times 2.905 \times 1.475}\]Calculate the multiplication inside the square root:\[A = \sqrt{6.695 \times 2.315 \times 2.905 \times 1.475} \approx \sqrt{66.171}\].Finally, calculate the square root:\[A \approx 8.135\].

Step 6: Round the Answer

Round the area to three significant digits: \(A \approx 8.14 \text{ square feet}\).

Key Concepts

Understanding the Semi-Perimeter

The semi-perimeter of a triangle is a helpful intermediary value used in Heron's formula, which calculates the area of a triangle when we know all three side lengths. Simply put, the semi-perimeter is half of the triangle's total perimeter. To find the semi-perimeter, add up the lengths of all three sides of the triangle, then divide that sum by two. For the given triangle, this calculation for the semi-perimeter, denoted as \( s \), looks like this:- Add the sides: \( a + b + c = 4.38 + 3.79 + 5.22 \)- Divide by 2 to get \( s \), so: \( s = \frac{4.38 + 3.79 + 5.22}{2} = 6.695 \)The semi-perimeter simplifies the process of using Heron's formula, which requires knowing this value before calculating the triangle's area.

Calculating Triangle Area with Heron's Formula

Heron's formula provides a neat solution for determining the area of a triangle when its side lengths are known. This method becomes particularly useful when the triangle is not a right triangle, ie., when simple trigonometric methods cannot be applied. Heron's formula states that the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) can be calculated using:- \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] Where \( s \) represents the semi-perimeter. Following through our example:

• First, \( s - a = 6.695 - 4.38 = 2.315 \)
• Then, \( s - b = 6.695 - 3.79 = 2.905 \)
• Finally, \( s - c = 6.695 - 5.22 = 1.475 \)

Plug these values back into Heron's formula:- \[ A = \sqrt{6.695 \times 2.315 \times 2.905 \times 1.475} \approx \sqrt{66.171} \] Ultimately, the area approximates to \( \sqrt{66.171} \approx 8.135 \). This result then needs to be rounded to an appropriate number of significant digits.

The Importance of Significant Digits

When reporting any calculated value, especially in measurements and geometry, maintaining a correct level of precision is vital. Significant digits represent the precision or certainty of your numbers and are commonly used to ensure consistent results. They help convey how accurately the measurement was taken or, in mathematical operations, what the practical certainty in a result is. Let's break this down with the triangle area from before:- We initially computed the area as \( 8.135 \) square feet.- However, the problem asks us to round to three significant digits.To round to three significant digits:- Start from the first non-zero digit, which here, in 8.135, are "8", "1", and "3".This results in a rounded area of \( 8.14 \) square feet. Significant digits ensure that the level of precision mirrors the reliability of the measurements and results provided, ensuring clarity and uniformity in your calculations.