Problem 1 - Basic Engineering Mathematics

Question

The area \(A\) of a triangle is given by \(A=\frac{1}{2} b h\). The base \(b\) when measured is found to be \(3.26 \mathrm{~cm}\), and the perpendicular height \(h\) is \(7.5 \mathrm{~cm}\). Determine the area of the triangle.

Step-by-Step Solution

Answer

The area of the triangle is 12.225 cm².

Step 1: Identify Given Values

First, we need to identify the values given in the problem. We know that the base of the triangle \(b\) is \(3.26\) cm and the height \(h\) is \(7.5\) cm.

Step 2: Write Down the Area Formula

The formula to calculate the area \(A\) of a triangle is given by \(A = \frac{1}{2} b h\). We will use this formula to calculate the area.

Step 3: Substitute the Given Values

Substitute the given values of \(b\) and \(h\) into the formula: \[ A = \frac{1}{2} \times 3.26 \times 7.5 \].

Step 4: Calculate the Area

First, compute the multiplication inside the formula: \(3.26 \times 7.5 = 24.45\). Then, calculate the area using the formula: \(A = \frac{1}{2} \times 24.45 = 12.225\).

Step 5: State the Final Result

The area of the triangle is \(12.225\) cm². Therefore, the triangle's area is approximately \(12.23\) cm² when rounded to two decimal places.

Key Concepts

Understanding Mathematics Behind Area of a Triangle

In mathematics, the area of a triangle is a fundamental concept that helps us understand the size of the space enclosed within a triangle's boundaries. It is pivotal in various applications, from architectural designs to geographic mapping. To find the area, the universally applied formula is \( A = \frac{1}{2} b h \), where \( b \) represents the base, and \( h \) stands for the height of the triangle. This formula simplifies the calculation by using the base and the perpendicular height, offering an efficient way to determine the area. By breaking down the problem into manageable steps, we can apply this formula effectively to solve real-world problems. Recognizing and applying this formula is a key step in various mathematical explorations and analyses.

Exploring Geometry Concepts in Triangles

Geometry deals with the shapes, sizes, and properties of figures, with triangles being one of the most foundational shapes. Understanding the geometry of triangles involves identifying key elements such as the base and height, both crucial for calculating the area. The base of a triangle can be any one of its sides, depending on which side you choose to base your height upon. The height is always measured as the perpendicular (right angle) distance from this base to the opposite vertex. This perpendicular measurement ensures accuracy in determining how much space the triangle occupies. By visualizing and identifying the correct base and height, one can comprehensively grasp the concepts involved in calculating the triangle's area. This understanding not only aids in solving problems but also enhances spatial reasoning skills.

Problem Solving with Triangle Area Calculations

Problem solving in mathematics often involves a systematic approach to identify, understand, and solve a given problem. When determining the area of a triangle, following these steps ensures that the solution is both correct and easy to reach.

• First, list all given values, like the base \( b = 3.26 \) cm and the height \( h = 7.5 \) cm as seen in the problem.
• Next, write down the area formula \( A = \frac{1}{2} b h \) to keep the method clear.
• Substitute the known values into the formula and solve: \( A = \frac{1}{2} \times 3.26 \times 7.5 \).
• Perform the arithmetic operations: First multiply \( 3.26 \times 7.5 = 24.45 \) and then find the area, resulting in \( A = 12.225 \) cm².

This strategic approach not only provides the solution but also instills a habit of logical reasoning and critical thinking. These skills are indispensable across different levels of mathematics and other analytical fields.