Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 7, give exact answers, not numerical approximations. A rectangle is 3 meters long and 2 meters high. How long is the diagonal?

Step-by-Step Solution

Answer

The length of the diagonal is \(\sqrt{13}\) meters.

Step 1: Identify the Given Information

The given information is the length and width of a rectangle. The length is 3 meters and the width is 2 meters.

Step 2: Write Down the Pythagorean Theorem

The Pythagorean theorem is \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the sides of the right triangle, and \(c\) is the hypotenuse.

Step 3: Substitute the Given Values into the Pythagorean Theorem

Substituting the given values into the equation becomes \(3^2 + 2^2 = c^2\).

Step 4: Solve for \(c\)

So, \(c^2 = 9 + 4 = 13\). Taking the square root of both sides, the length of the diagonal is \(\sqrt{13}\) meters.

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics that explores the properties and relationships of shapes, sizes, and figures in space. It includes the study of points, lines, surfaces, angles, and solids. In this exercise, we're dealing with a rectangle, which is a fundamental shape in geometry.

A rectangle is a quadrilateral with four right angles and opposite sides that are equal in length. Because of its predictable shape, a rectangle makes it easy to apply geometric principles such as the Pythagorean Theorem for calculations. Understanding these principles is crucial, as they help us determine unknown values such as the diagonal in this case. When working with rectangles:

• The diagonals are always equal.
• They intersect each other at the center of the rectangle.
• The diagonals can be calculated when the rectangle's sides are known, by using the Pythagorean Theorem.

These characteristics make rectangles an exceptional subject for learning fundamental geometric concepts.

Exploring Right Triangles

A right triangle is a special type of triangle that includes a 90-degree angle. It consists of three sides: the hypotenuse, which is the longest side opposing the right angle, and two legs, which are the triangle's other sides. In geometry, right triangles are essential because they introduce the Pythagorean Theorem.

In this particular exercise, the rectangle forms a right triangle when divided by its diagonal. Here, the diagonal acts as the hypotenuse, while the rectangle's length and height represent the two legs of the triangle. Key points about right triangles include:

• The Pythagorean Theorem applies exclusively to right triangles, allowing calculations of one side if the other two are known.
• Right triangles are used in various real-world applications, such as construction and physics.

Understanding right triangles and their properties is foundational in Euclidean geometry and is applied in many practical scenarios.

Diagonal Calculation Using Pythagorean Theorem

The diagonal of a rectangle is essentially the same as finding the hypotenuse of a right triangle formed by the rectangle's sides. To find this diagonal, we can utilize the Pythagorean Theorem, which is a key formula in geometry:

The Pythagorean Theorem is stated as:\[a^2 + b^2 = c^2\]Where:- \(a\) and \(b\) refer to the two shorter sides of the triangle (the rectangle's length and width).- \(c\) denotes the diagonal (or hypotenuse) of the triangle.

To calculate the diagonal for our rectangle with a length of 3 meters and a height of 2 meters:

• Substitute the known values into the formula: \(3^2 + 2^2 = c^2\).
• Calculate: \(9 + 4 = 13\).
• Find \(c\) by taking the square root of 13: \(c = \sqrt{13}\).

This final answer, \(\sqrt{13}\) meters, is the exact length of the rectangle's diagonal. Understanding this process is crucial in various fields including architecture and engineering, where precise measurements are necessary.