Problem 16 - Basic Technical Mathematics with Calculus

Question

Find the area of each triangle. Equilateral triangle of sides 3200 yd

Step-by-Step Solution

Answer

The area is approximately 4433920 square yards.

Step 1: Recall the Formula for the Area of an Equilateral Triangle

The formula to find the area of an equilateral triangle with side length \( s \) is given by \[ A = \frac{\sqrt{3}}{4} s^2 \]. This is a specific case of the general triangle area formula and takes advantage of the symmetry of equilateral triangles.

Step 2: Substitute the Given Side Length into the Formula

The side length of the given equilateral triangle is 3200 yards. Substitute \( s = 3200 \) into the formula: \[ A = \frac{\sqrt{3}}{4} \times 3200^2 \].

Step 3: Compute the Square of the Side Length

Calculate \( 3200^2 \): \( 3200 \times 3200 = 10240000 \).

Step 4: Simplify the Formula with Values Substituted

Substitute \( 10240000 \) back into the formula: \[ A = \frac{\sqrt{3}}{4} \times 10240000 \].

Step 5: Calculate the Area

Calculate \( \frac{\sqrt{3}}{4} \): Multiply \( \frac{\sqrt{3}}{4} \) by \( 10240000 \). Since \( \sqrt{3} \approx 1.732 \), we have \( \frac{\sqrt{3}}{4} \approx 0.433 \). Multiply \( 0.433 \times 10240000 = 4433920 \).

Step 6: Final Result

The area of the equilateral triangle with side length 3200 yards is approximately \( 4433920 \) square yards.

Key Concepts

geometry

Geometry is a branch of mathematics that focuses on the study of shapes, sizes, and properties of space. In geometry, we explore various figures and their relationships with one another. This includes understanding how different forms such as triangles, circles, and polygons are structured and how they interact.
The fundamental building blocks of geometry include points, lines, and angles. From these, we can develop complex shapes and solve intricate problems. For triangles, one of the most interesting types to explore is the equilateral triangle. All sides and angles in an equilateral triangle are equal, making it a fascinating subject of study. These unique features lead to specific formulas that help us compute various properties, like the area, with precision.

triangle area formula

Calculating the area of a triangle is a central concept in geometry. The area can tell us how much space is covered by the triangle on a plane. For a general triangle, the area is found using the formula:

• Area = \( \frac{1}{2} imes \text{base} imes \text{height} \)

However, for an equilateral triangle, which has equal-length sides and equal angles, the formula becomes more straightforward. The area of an equilateral triangle can be calculated using the formula:

• Area = \( \frac{\sqrt{3}}{4} s^2 \)

where \( s \) represents the length of one side. This formula is derived knowing that in an equilateral triangle, the height forms two 30-60-90 right triangles within. Calculating the area offers insight into the nature of symmetry found in equilateral triangles, highlighting how formulas are adapted to fit specific conditions.

mathematics

Mathematics, as a field, encompasses a wide range of topics, including number theory, algebra, and geometry. Each area of mathematics explores different aspects of numbers and shapes, providing tools to solve real-world problems. Within the scope of mathematics, geometric figures like the equilateral triangle are studied in detail because of their regular properties and predictable behaviors.
Understanding the mathematics behind equilateral triangles, and their area involves combining algebraic formulas and geometric insights. By using specific formulas like \( A = \frac{\sqrt{3}}{4} s^2 \), mathematicians can solve for properties of these unique shapes efficiently.
Through mathematical reasoning and calculations, concepts like the equilateral triangle's area formula not only deepen our understanding of geometry but also sharpen problem-solving skills that are applicable in various scientific fields.