### Estimating the Area of Virginia Using Heron's Formula#### Heron's FormulaTo find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula:\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]Here, $ a $, $ b $, and $ c $ are the lengths of the sides, and $ s $ is the semi-perimeter of the triangle, which is half of the perimeter:\[ s = \frac{a + b + c}{2} \]#### Example CalculationTo estimate the area of Virginia, assume it resembles a triangle with the following side lengths:- Side $ a = 370 \, \text{mi} $- Side $ b = 220 \, \text{mi} $- Side $ c = 430 \, \text{mi} $##### Step-by-Step Solution1. **Calculate the Semi-Perimeter**: \[ s = \frac{a + b + c}{2} = \frac{370 + 220 + 430}{2} = 510 \, \text{mi} \]2. **Apply Heron's Formula**: \[ A = \sqrt{510(510 - 370)(510 - 220)(510 - 430)} \]3. **Simplify the Expression**: \[ A = \sqrt{510 \cdot 140 \cdot 290 \cdot 80} \]4. **Calculate the Final Area** (skipping intermediate calculations for brevity): \[ A \approx 43,011 \, \text{mi}^2 \]So, the area of Virginia (to the nearest square mile) is approximately **43,011 square miles**.#### Diagram ExplanationThe diagram provided illustrates a triangle with sides labeled as follows:- The base $ a $ is 430 miles.- One side $ b $ is 370 miles.- The other side $ c $ is 220 miles.This visual aids in understanding how the dimensions of the state approximate those of a triangle for the area calculation.

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Description: ### Estimating the Area of Virginia Using Heron's Formula#### Heron's FormulaTo find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula:\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]Here, $ a $, $ b $, and

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The formula A = [((s - a)(s - b)(s - c)]/2 gives the area of a triangle with sides of length a, b, and c, where s is one-half of the perimeter. Estimate the area of Virginia (to the nearest square mile) using the data given below. A = |mi2 370 mi 430 mi 220 mi

The formula A = [((s - a)(s - b)(s - c)]/2 gives the area of a triangle with sides of length a, b, and c, where s is one-half of the perimeter. Estimate the area of Virginia (to the nearest square mile) using the data given below. A = |mi2 370 mi 430 mi 220 mi

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Estimating the Area of Virginia Using Heron's Formula

#### Heron's Formula
To find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula:

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

Here, $ a $, $ b $, and $ c $ are the lengths of the sides, and $ s $ is the semi-perimeter of the triangle, which is half of the perimeter:

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

#### Example Calculation
To estimate the area of Virginia, assume it resembles a triangle with the following side lengths:

- Side $ a = 370 \, \text{mi} $
- Side $ b = 220 \, \text{mi} $
- Side $ c = 430 \, \text{mi} $

##### Step-by-Step Solution
1. **Calculate the Semi-Perimeter**:
\[
s = \frac{a + b + c}{2} = \frac{370 + 220 + 430}{2} = 510 \, \text{mi}
\]

2. **Apply Heron's Formula**:
\[
A = \sqrt{510(510 - 370)(510 - 220)(510 - 430)}
\]

3. **Simplify the Expression**:
\[
A = \sqrt{510 \cdot 140 \cdot 290 \cdot 80}
\]

4. **Calculate the Final Area** (skipping intermediate calculations for brevity):
\[
A \approx 43,011 \, \text{mi}^2
\]

So, the area of Virginia (to the nearest square mile) is approximately **43,011 square miles**.

#### Diagram Explanation
The diagram provided illustrates a triangle with sides labeled as follows:

- The base $ a $ is 430 miles.
- One side $ b $ is 370 miles.
- The other side $ c $ is 220 miles.

This visual aids in understanding how the dimensions of the state approximate those of a triangle for the area calculation.

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