Question

The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of the land is \(\$ 3\) per square foot, how much does the lot cost?

Short answer

The lot costs \$5625.

Step-by-step solution

- Identify the Sides of the Triangle

The sides of the triangular lot are given as 100 feet, 50 feet, and 75 feet.

- Use Heron's Formula to Find the Area

Firstly, calculate the semi-perimeter of the triangle:enote the semi-perimeter as s, where \[ s = \frac{a + b + c}{2} \]Given: a = 100, b = 50, c = 75\[ s = \frac{100 + 50 + 75}{2} = 112.5 \]Using Heron's formula:e area (A) is given by:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]\[ A = \sqrt{112.5(112.5 - 100)(112.5 - 50)(112.5 - 75)} \]\[ A = \sqrt{112.5 \cdot 12.5 \cdot 62.5 \cdot 37.5} \]

- Calculate the Specific Area

Compute the product inside the square root:\[ A = \sqrt{112.5 \times 12.5 \times 62.5 \times 37.5} \]This computation gives:\[ A \approx 1875 \text{ square feet} \]

- Determine the Cost of the Lot

Given that the cost per square foot is \$3, calculate the total cost:\[ \text{Total Cost} = 3 \text{ dollars/foot}^2 \times 1875 \text{ square feet} \]\[ \text{Total Cost} = 5625 \text{ dollars} \]

Key concepts

Triangular Lot

Understanding a triangular lot is key to solving the problem. A triangular lot has three sides and three angles. In this exercise, the sides are given as 100 feet, 50 feet, and 75 feet. Think of the lot as a large piece of land shaped like a triangle. The sides of the triangle are the boundaries of this land. By identifying the dimensions of the lot, we can use formulas to find other important measurements, like the area. This step is crucial for further calculations.

Semi-perimeter

The semi-perimeter is half of the triangle's total perimeter. To use Heron's formula for finding the area, we first need to calculate the semi-perimeter. For a triangle with sides a, b, and c, the semi-perimeter (s) is calculated as:

Area Calculation

Calculating the area of a triangular lot involves using Heron's formula. Heron's formula is a way of finding the area when you know the lengths of all three sides. After calculating the semi-perimeter (s), the area (A) can be computed using the formula:

Cost Estimation

Once we have the area of the triangular lot, we can calculate the cost. If the price per square foot is given, multiplying it by the total area gives the total cost. In this exercise, the cost per square foot is $3. Therefore, the formula to calculate the total cost is: