Problem 51 - 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 \(\mathrm{ft}\) ?

Step-by-Step Solution

Answer

The lot's value is approximately $163,859.60.

Step 1: Identify the Method

Since we have the lengths of the three sides of the triangle, we can use Heron's formula to find the area of the triangular lot, which allows calculating the value of the land.

Step 2: Calculate the Semi-perimeter

The semi-perimeter of a triangle is calculated as half the sum of its sides. Compute the semi-perimeter, denoted as \(s\):\[ s = \frac{112 + 148 + 190}{2} = 225 \]

Step 3: Apply Heron's Formula

Heron's Formula to find the area \(A\) of a triangle is: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \(a = 112\), \(b = 148\), and \(c = 190\). Substitute the values: \[ A = \sqrt{225(225-112)(225-148)(225-190)} \]

Step 4: Simplify Inside the Square Root

Calculate the expressions: \(225 - 112 = 113\), \(225 - 148 = 77\), \(225 - 190 = 35\). Substitute these into Heron's formula: \[ A = \sqrt{225 \times 113 \times 77 \times 35} \]

Step 5: Compute the Area

Using a calculator, compute: \[ A = \sqrt{225 \times 113 \times 77 \times 35} \approx 8192.98 \text{ square feet}\]

Step 6: Calculate the Land Value

Multiply the area by the value per square foot: \[ \text{Value} = 8192.98 \times 20 = 163,859.60\]

Key Concepts

Semi-perimeter

Calculating the semi-perimeter is an essential first step when using Heron's formula to find the area of a triangle. The semi-perimeter of a triangle is half the sum of its three sides. It is denoted by the letter \( s \). Here’s how you can calculate it:

• Add up the lengths of all sides of the triangle. In our exercise, the sides are 112 ft, 148 ft, and 190 ft.
• The sum of these sides is 450 ft.
• Divide the sum by 2 to get the semi-perimeter. Hence, \( s = \frac{450}{2} = 225 \).

This value is crucial as it simplifies the computation of the area using Heron's formula. Understanding and calculating the semi-perimeter correctly ensures accurate calculations of more complex aspects such as the area of the triangle.

Triangle Area Calculation

Heron's formula is a smart and efficient method to calculate the area of any triangle when the lengths of all three sides are known. The formula is as follows:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle. Let’s break it down with our exercise:

• We have already determined that the semi-perimeter \( s = 225 \).
• Substitute the values of the sides: \( a = 112 \), \( b = 148 \), and \( c = 190 \).
• Compute each term: \( 225 - 112 = 113 \), \( 225 - 148 = 77 \), and \( 225 - 190 = 35 \).
• Plug these values into Heron's formula: \( A = \sqrt{225 \times 113 \times 77 \times 35} \).
• Calculate the result: The area \( A \approx 8192.98 \text{ square feet} \).

Understanding how to apply Heron's formula takes complex geometry tasks and makes them approachable, offering precise results with simple arithmetic.

Land Valuation

Valuing land effectively requires knowing the area of the land, especially when dealing with irregular shapes like triangles. Once we have calculated the area using Heron's formula, we can find the monetary value by multiplying the area by the price per square foot.

• First, ensure you have the area, which we've found to be approximately 8192.98 square feet.
• The value of land per square foot is given as $20.
• Multiply: \( \text{Value} = 8192.98 \times 20 \).
• Calculate the land's total value, yielding \( \text{Value} \approx 163,859.60 \text{ dollars} \).

This is a practical application of mathematical principles to real-world scenarios like land pricing, making geometric calculations incredibly useful in fields such as real estate and finance.