Land in downtown Columbia is valued at $20 a square foot. What is the value of a triangular lot with sides of lengths 112, 146, and 186 ft? (Round your answer to the nearest dollar.)


To determine the value of the triangular lot, we first need to calculate its area using Heron's formula. Once we have the area, we can multiply it by the value per square foot to find the total value.

### Step 1: Use Heron's Formula to Find the Area

Heron's formula states that the area $ A $ of a triangle with sides of lengths $ a $, $ b $, and $ c $ is given by:

\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]

where $ s $ is the semi-perimeter of the triangle:

\[
s = \frac{a + b + c}{2}
\]

Given side lengths:

- $ a = 112 $ ft
- $ b = 146 $ ft
- $ c = 186 $ ft

### Step 2: Calculate the Semi-perimeter

\[
s = \frac{112 + 146 + 186}{2} = \frac{444}{2} = 222 \text{ ft}
\]

### Step 3: Calculate the Area

\[
A = \sqrt{222(222 - 112)(222 - 146)(222 - 186)}
\]

Simplifying the terms inside the square root:

\[
A = \sqrt{222 \times 110 \times 76 \times 36}
\]

Let's calculate this value step-by-step.The area of the triangular lot is approximately $ 8173.93 $ square feet.

### Step 4: Calculate the Value of the Lot

The value of the land is $ 20 $ dollars per square foot. Thus, the total value $ V $ of the lot is:

\[
V = 8173.93 \times 20
\]

Let's calculate this value and round it to the nearest dollar.The value of the triangular lot is approximately **$163,479**.

Would you like more details or have any questions?

Here are some related questions to explore:

1. How does Heron's formula work for different triangle types?
2. What happens if one side length changes? How does that affect the area and value?
3. How do you determine the value per square foot in different locations?
4. What are some alternative methods to calculate the area of irregular shapes?
5. How do changes in land value impact real estate investments?

**Tip:** When dealing with irregular shapes or lots, using precise area calculation methods like Heron's formula can help ensure accurate valuations.

Land in downtown Columbia is valued at $20 a square foot. What is the value of a triangular lot with sides of lengths 112, 146, and 186 ft? (Round your answer to the nearest dollar.)

Discover how to calculate the value of a triangular lot in downtown Columbia using Heron's formula. This problem involves determining the area of a triangle with sides 112, 146, and 186 feet, and then finding the total land value at $20 per square foot. Ideal for students in Grades 10-12.

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