Problem 47 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. The sail of a sailboat is in the shape of a right triangle with sides of \(8.0 \mathrm{ft}, 15 \mathrm{ft},\) and \(17 \mathrm{ft} .\) What is the area of the sail?

Step-by-Step Solution

Answer

The area of the sail is 60 square feet.

Step 1: Identify the Triangle Type

The problem states the sides of the sail are 8 ft, 15 ft, and 17 ft. Check whether it is a right triangle by applying the Pythagorean theorem.\[ a^2 + b^2 = c^2 \] where \( c = 17 \) ft (the longest side), \( a = 8 \) ft, and \( b = 15 \) ft. Compute: \[ 8^2 + 15^2 = 64 + 225 = 289 \] which matches \( 17^2 = 289 \). Thus, it is a right triangle.

Step 2: Use the Formula for Area of a Right Triangle

The area \(A\) of a right triangle can be found using the formula \[ A = \frac{1}{2} imes ext{base} imes ext{height} \]. For this sail, the base could be 8 ft and the height 15 ft (or vice versa).

Step 3: Calculate the Area

Substitute the base and height into the formula:\[ A = \frac{1}{2} imes 8 imes 15 \]. Calculate the area: \[ A = 60 \text{ square feet} \].

Key Concepts

Right Triangles

In geometry, a right triangle is a special type of triangle that contains one angle measuring exactly 90 degrees. This right angle creates a distinctive triangle shape where the side opposite the right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the legs of the triangle.

Right triangles are fundamental in geometry due to their unique properties and their widespread applicability in different problem-solving scenarios.

• They have a one-to-one correspondence to right angles, which are always present in such triangles.
• The two legs meet at the right angle and can vary in length, but only under the condition that one side is always longer than both legs, which is the hypotenuse.
• Right triangles allow for the application of various mathematical principles such as the Pythagorean theorem, trigonometric ratios, and more.

Whenever you're working with or identifying right triangles, remember the significance of the right angle, as it is integral to defining the triangle's properties and functionality.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry. It's used to determine whether a triangle is a right triangle and is expressed with the formula:
\[ a^2 + b^2 = c^2\]
Here, \(a\) and \(b\) are the lengths of the right triangle's legs, while \(c\) is the length of the hypotenuse. The theorem states that the square of the hypotenuse's length is equivalent to the sum of the squares of the other two sides.

This theorem has a variety of real-world applications, helping to solve problems involving displacement, distance, and more.

• The formula can confirm if a given triangle is a right triangle by checking if the theorem holds true.
• It provides a mathematical foundation to calculate distances without directly measuring them.
• The verification of the theorem involves having three specific numerical values where the squared sum of two equals the squared third.

Using the theorem makes it straightforward to identify if a triangle is right-angled, thus facilitating further geometric calculations.

Area Calculation

Calculating the area of a right triangle is straightforward once you have verified that it is indeed a right triangle. The formula for the area of a right triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here are the steps to follow:

• Identify which sides serve as the base and height, often the two legs of the right triangle.
• Substitute these values into the formula.
• Solve for the area.

For the exercise mentioned, with one leg as the base (8 ft) and the other as the height (15 ft), you substitute these values into the equation, yielding:
\[ A = \frac{1}{2} \times 8 \times 15 = 60 \]
Thus, the area of this right triangle-shaped sailboat sail is \(60\) square feet. This calculation method provides an easy way to measure the effectiveness of space usage in various practical applications, such as construction, navigation, and design.