Math Problem Statement

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.)

Solution

To find the value of the triangular lot, we first need to calculate its area and then multiply it by the value per square foot.

Step 1: Use Heron's formula to find the area of the triangle.

Heron's formula is used to find the area of a triangle when the lengths of all three sides are known. The formula is:

A=s(sa)(sb)(sc)A = \sqrt{s(s-a)(s-b)(s-c)}

where:

  • AA is the area,
  • aa, bb, and cc are the lengths of the sides of the triangle,
  • ss is the semi-perimeter, calculated as:

s=a+b+c2s = \frac{a + b + c}{2}

Step 2: Calculate the semi-perimeter.

For the given triangle with side lengths a=112a = 112, b=146b = 146, and c=186c = 186:

s=112+146+1862=4442=222s = \frac{112 + 146 + 186}{2} = \frac{444}{2} = 222

Step 3: Apply Heron's formula.

Substitute the values of ss, aa, bb, and cc into the formula:

A=222(222112)(222146)(222186)A = \sqrt{222(222 - 112)(222 - 146)(222 - 186)} A=222(110)(76)(36)A = \sqrt{222(110)(76)(36)}

Now, let's calculate this.

The area of the triangular lot is approximately 8,174 square feet.

Step 4: Calculate the value of the lot.

The value of the land is 2020 dollars per square foot. Therefore, the total value of the lot is:

Value=20×8174=163,480 dollars\text{Value} = 20 \times 8174 = 163,480 \text{ dollars}

Thus, the value of the triangular lot is approximately $163,480.

Would you like more details or have any questions?

Here are 5 related questions:

  1. How does Heron's formula apply to other types of triangles?
  2. How would the problem change if the lot were valued per acre instead of square feet?
  3. Can you calculate the perimeter of this triangular lot?
  4. What if the land value changes? How would it affect the total price?
  5. How do you calculate the area of irregularly shaped lots?

Tip: Heron's formula works for any triangle, not just right-angled ones, making it versatile for real-world applications.

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Math Problem Analysis

Mathematical Concepts

Geometry
Area of Triangles
Heron's Formula

Formulas

Heron's formula: A = sqrt(s(s-a)(s-b)(s-c))
Semi-perimeter: s = (a + b + c) / 2

Theorems

Heron's Theorem

Suitable Grade Level

Grades 9-12