Question

The triangular sail on a boat has a height of \(4.1 \mathrm{~m}\) and a base of \(6.15 \mathrm{~m}\). What is the area of the sail? Recall that the area of a triangle is given by area \(=\frac{1}{2}\) (base)(height).

Short answer

The area of the sail is \(12.61 \, \mathrm{m^2}\).

Step-by-step solution

Identify the Formula

The problem requires us to find the area of a triangle, which is given by the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).

Substitute Known Values

We know that the base of the triangle is \(6.15 \, \mathrm{m}\) and the height is \(4.1 \, \mathrm{m}\). Substitute these values into the area formula: \( \text{Area} = \frac{1}{2} \times 6.15 \times 4.1 \).

Calculate the Product

First, calculate the product of the base and height: \( 6.15 \times 4.1 = 25.215 \).

Apply the Multiplicative Factor

Apply the \(\frac{1}{2}\) factor to the product from the previous step: \( \text{Area} = \frac{1}{2} \times 25.215 = 12.6075 \).

State the Final Answer

Round the area to two decimal places for simplicity, which is customary for measurements: the final area of the sail is \(12.61 \, \mathrm{m^2}\).

Key concepts

Geometry

Geometry is an essential branch of mathematics that involves the study of shapes, sizes, and the properties of space. When we look at shapes like triangles, geometry helps us understand their structure and helps solve related problems.
A triangle is one of the fundamental shapes studied in geometry. It is a three-sided polygon, and the sum of its interior angles is always 180 degrees. When calculating the area of such shapes, geometry provides the tools and formulas needed for precise calculations.
In the case of the triangular sail, knowing how to apply geometrical principles to determine its area is crucial. The formula for the area of a triangle, in this context, is particularly useful. It is expressed as:

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

Understanding and utilizing this formula means we can easily pinpoint how much material is needed to cover the sail, which transforms geometry from theoretical concepts into practical applications.

Triangular Sail

Triangular sails are a common feature in sailing that harness the power of the wind efficiently. These sails are not only important in marine navigation but also serve as a fantastic practical example for applying mathematical concepts.
When determining the size or area of a triangular sail, you’re essentially figuring out how much surface area will catch the wind. The sail's height and base are critical measurements for this determination.
Here’s why these measurements are important:

• The **base** is the bottom edge of the triangle, often measured in a straight line from one end to the other.
• The **height** is the perpendicular measurement from the base to the opposite vertex (the top point) of the triangle.

Together, these measurements allow us to compute the sail's area by plugging them into the area formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). This approach simplifies what might seem like a complex geometry problem into manageable steps.

Mathematics Problem Solving

Mathematics problem solving is a critical skill that can be developed with practice and understanding of key concepts. Solving a problem like finding the area of a triangular sail involves a series of logical steps that build on each other.
Here's a typical approach you can take:

• **Identify the formula**: Recognize the correct formula needed for the problem. In the case of a triangle's area, use \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
• **Substitute known values**: Take the values provided in the problem, such as the base and height, and substitute them into the formula.
• **Perform calculations**: Execute the necessary arithmetic operations, first multiplying the base by the height, then by \( \frac{1}{2} \).
• **State the final answer**: After performing calculations, consider rounding the answer if necessary, and present it clearly.

This structured method is valuable not just for solving problems related to sails but for a wide range of mathematical challenges. By practicing this approach, you cultivate a methodical way of thinking that can greatly enhance problem-solving efficiency.