Problem 53 - Algebra and Trigonometry

Question

Land in downtown Columbia is valued at \(\$ 20\) a square foot. What is the value of a triangular lot with sides of lengths 112, 148, and 190 ft?

Step-by-Step Solution

Answer

The lot is valued at approximately \$165,359.20.

Step 1: Identify the Formula to Use

We will use Heron's formula to find the area of the triangle. Heron's formula is appropriate because we know the lengths of all the sides of the triangle. The formula is \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( s \) is the semi-perimeter defined as \( s = \frac{a+b+c}{2} \).

Step 2: Calculate the Semi-perimeter

Sum up the sides and divide by 2 to find the semi-perimeter: \( s = \frac{112 + 148 + 190}{2} \). Calculate this to get \( s = 225 \) ft.

Step 3: Apply Heron's Formula

Substitute into Heron's formula: \( A = \sqrt{225(225-112)(225-148)(225-190)} \). Calculate each term inside the square root: \( 225-112 = 113 \), \( 225-148 = 77 \), \( 225-190 = 35 \). This gives \( A = \sqrt{225 \times 113 \times 77 \times 35} \).

Step 4: Calculate the Area

Calculate the expression under the square root: \( 225 \times 113 \times 77 \times 35 = 68375775 \). Take the square root of this number to find the area: \( A \approx 8267.96 \) square feet.

Step 5: Calculate the Value of the Lot

The value of the lot is based on \$20 per square foot. Multiply the area by the value per square foot \( Value = 8267.96 \times 20 \). This gives \( Value = 165359.2 \).

Key Concepts

Understanding Semi-Perimeter

When solving problems related to triangles, especially when you have the lengths of all three sides, the concept of the semi-perimeter becomes handy. The semi-perimeter of a triangle is half the sum of its sides. It is a crucial component of Heron's Formula for calculating the area of a triangle. In our example, with side lengths 112 feet, 148 feet, and 190 feet, here’s how you find the semi-perimeter:

• Add up all the side lengths: \( 112 + 148 + 190 = 450 \) ft.
• Divide this total by 2 to get the semi-perimeter: \( s = \frac{450}{2} = 225 \) ft.

This semi-perimeter value acts as an intermediary step in further calculations needed to find the triangle's area using Heron’s Formula.

Triangle Area Calculation with Heron's Formula

Heron's Formula provides a way to calculate the area of a triangle when you know the lengths of all its sides. This formula is particularly useful because it does not require you to know the height of the triangle. The formula is given by:\[A = \sqrt{s(s-a)(s-b)(s-c)}\]In this formula, \( s \) stands for the semi-perimeter, while \( a \), \( b \), and \( c \) are the side lengths of the triangle. Let’s apply this to a triangle with sides of 112, 148, and 190 ft:

• Using the semi-perimeter value: \( s = 225 \) ft, you substitute into the formula.
• You calculate: \( 225 - 112 = 113 \), \( 225 - 148 = 77 \), and \( 225 - 190 = 35 \).
• Insert these values into Heron's Formula to get: \( A = \sqrt{225 \times 113 \times 77 \times 35} \).
• Compute the product: \( 225 \times 113 \times 77 \times 35 = 68375775 \).
• Finally, take the square root to find the area: \( A \approx 8267.96 \) square feet.

This area measurement can then be used to determine other necessary values related to the triangle's property.

Calculating the Value of Land per Square Foot

Once you have calculated the area of a triangular piece of land, you can easily find its value if given a price per unit area, such as per square foot. In our example, the land in downtown Columbia is valued at $20 per square foot. Here is how you find the total value of the triangular lot:

• First, determine the total area of the triangle: approximately 8267.96 square feet, as calculated using Heron's Formula.
• Multiply this area by the price per square foot to find the total value: \( 8267.96 \times 20 = 165359.2 \) dollars.

This final step provides a practical application of the mathematical calculation, allowing landowners or businesses to assess the financial worth of a property accurately.