Problem 28 - 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. \(a=8.32 \mathrm{ft}, b=12.36 \mathrm{ft}, c=5.34 \mathrm{ft}\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 17.5 ft².

Step 1: Understand the Problem

We need to find the area of triangle \(ABC\) using the given side lengths \(a = 8.32\, \, \text{ft}\), \(b = 12.36\, \, \text{ft}\), and \(c = 5.34\, \, \text{ft}\). The side lengths suggest using Heron's formula for area.

Step 2: Calculate the Semiperimeter

First, compute the semiperimeter \(s\) of the triangle. The formula is:\[ s = \frac{a+b+c}{2} = \frac{8.32 + 12.36 + 5.34}{2} \]Calculate \(s\).

Step 3: Solve for the Semiperimeter

Perform the calculations:\[ s = \frac{8.32 + 12.36 + 5.34}{2} = \frac{26.02}{2} = 13.01\, \text{ft} \]Hence, the semiperimeter \(s = 13.01\, \, \text{ft}\).

Step 4: Use Heron's Formula

Using Heron's formula, the area \(A\) of the triangle is given by:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the values of \(s\), \(a\), \(b\), and \(c\) into the formula.

Step 5: Substitute and Calculate

Substitute the known values into Heron's formula:\[ A = \sqrt{13.01(13.01-8.32)(13.01-12.36)(13.01-5.34)} \]\[ A = \sqrt{13.01(4.69)(0.65)(7.67)} \]

Step 6: Compute Under the Square Root

Calculate the expression under the square root:\[ 13.01 \times 4.69 \times 0.65 \times 7.67 \approx 305.926 \]This is the product we need to find the square root of.

Step 7: Final Calculation

Find the square root of the computed product:\[ A = \sqrt{305.926} \approx 17.49 \]Thus, the area of the triangle is approximately \(17.49\, \, \text{ft}^2\).

Step 8: Round the Answer

Round your answer to three significant digits as required:The final area of triangle \(ABC\) is \(17.5\, \, \text{ft}^2\).

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle can sometimes seem daunting, but with the right method, it becomes quite manageable. For instance, one can use Heron's Formula, which doesn't require any angles or heights, only the lengths of the sides
. This makes it especially handy in scenarios where you know three sides, such as in triangle \(ABC\) with sides \(a=8.32\,\text{ft}\), \(b=12.36\,\text{ft}\), and \(c=5.34\,\text{ft}\).

• Firstly, compute the semiperimeter \(s\) of the triangle.
• Next, use Heron's formula \(A = \sqrt{s(s-a)(s-b)(s-c)}\).
• Substitute the values to find \(A\), the area of the triangle.

The beauty of Heron's Formula lies in its simplicity, allowing you to calculate the area directly from the side lengths. Remember, always use the semiperimeter to ease the computation before applying the full formula.

Semiperimeter Calculation

The semiperimeter is a crucial step in using Heron's Formula. It's essentially half of the triangle's perimeter. To find the semiperimeter \(s\) for a triangle, you simply add up all the side lengths and divide by 2 as follows:
\( s = \frac{a+b+c}{2} \)
For triangle \(ABC\), the calculation would be:
\( s = \frac{8.32 + 12.36 + 5.34}{2} = 13.01\,\text{ft} \)

• Add: Combine all three side lengths of the triangle.
• Divide: Take the total and divide by 2 to get \(s\).

This value of \(s\) becomes a fundamental part of the formula, simplifying the process of area calculation without the need for additional information like angles or heights.
Understanding this step ensures you're well on your way to mastering Heron's Formula.

Significant Figures in Mathematics

Significant figures are vital in mathematics because they indicate the precision of a measurement. When performing calculations, especially involving different operations, it’s important to manage significant figures correctly. In the context of our triangle area calculation, we've rounded the final answer to three significant digits. Here's how you do it:

• Identify the number of significant figures required. In this case, it was three.
• Look at the calculated value before rounding; for instance, our area was \(17.49\,\text{ft}^2\).
• Adjust by rounding to the specified number of figures. If the next figure is 5 or greater, round up the last significant digit.

Thus, \(17.49\,\text{ft}^2\) rounded to three significant figures becomes \(17.5\,\text{ft}^2\).
This ensures precision and clarity in communication, making sure results are as accurate as the measurements used to calculate them.
Practicing attention to significant figures is key in scientific calculations, avoiding misinterpretation of precision.