Problem 67 - Elementary and Intermediate Algebra

Question

The allowable length of a rectangular soccer field used for international adult matches can be from 100 to 110 meters and the width can be from 64 to 75 meters. a. Find the length of the diagonal of the field that has the minimum allowable length and minimum allowable width. Give an approximation to two decimal places. b. Find the length of the diagonal of the field that has the maximum allowable length and maximum allowable width. Give the exact answer and an approximation to two decimal places.

Step-by-Step Solution

Answer

Minimum diagonal: 118.71 meters; Maximum diagonal: approximately 133.19 meters.

Step 1: Determine Dimensions for Minimum Diagonal

To find the length of the diagonal of the field with the minimum allowable dimensions, use the minimum values for both length and width. Thus, the minimum length is 100 meters, and the minimum width is 64 meters.

Step 2: Calculate Minimum Diagonal Using Pythagorean Theorem

The diagonal of a rectangle can be calculated using the Pythagorean theorem: \ \[ d = \sqrt{L^2 + W^2} \] \ Plug in the minimum values for length (\(L = 100\)) and width (\(W = 64\)): \ \[ d = \sqrt{100^2 + 64^2} \] \ \[ d = \sqrt{10000 + 4096} \] \ \[ d = \sqrt{14096} \] \ \[ d \approx 118.71 \] meters. Therefore, the approximate length of the minimum diagonal is 118.71 meters.

Step 3: Determine Dimensions for Maximum Diagonal

For the maximum diagonal, use the maximum allowable values for length and width. Thus, the maximum length is 110 meters and the maximum width is 75 meters.

Step 4: Calculate Maximum Diagonal Using Pythagorean Theorem

Using the same formula: \ \[ d = \sqrt{L^2 + W^2} \] \ Plug in the maximum values for length (\(L = 110\)) and width (\(W = 75\)): \ \[ d = \sqrt{110^2 + 75^2} \] \ \[ d = \sqrt{12100 + 5625} \] \ \[ d = \sqrt{17725} \] \ The length of the maximum diagonal is exactly \( \sqrt{17725} \), and approximately: \ \[ d \approx 133.19 \] meters.

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with sizes, shapes, and the properties of space. In the context of flat surfaces, geometry involves understanding figures like triangles, rectangles, and circles. Geometry is not just a school subject; it is a fundamental tool used in various aspects of daily life and different professions. Architects use geometry to design buildings. Engineers apply it in drafting plans. Even artists rely on geometric principles for creating visual balance. In terms of mathematics, geometry often involves calculations and the use of special theorems to solve problems related to the dimensions and angles of different shapes.
Geometry provides the foundation for analyzing a rectangular field. One of its key principles involves understanding how different shapes can fit together or influence each other. You will often need to use theorems and formulas like the Pythagorean Theorem, which connects the sides of right triangles. This theorem is pivotal in calculating distances in various geometric contexts, such as determining the diagonal of a rectangle.

Rectangular Field Dimensions

Dimensions are essential to describe any rectangle. In the case of a rectangular soccer field, dimensions are described in terms of length and width. The length is the longer side, while the width is the shorter side. Knowing these dimensions allows you to calculate other characteristics of the field, like area and perimeter. The area is calculated by multiplying the length by the width, giving you the total space within the boundary of the field.
Specifically, in the context of a soccer field used for international matches, there are specified ranges for both the length and width. The length can vary between 100 to 110 meters, and the width can range from 64 to 75 meters. To find the field's diagonal, these are the key numbers you need to use in mathematical calculations.

• The length is the greater of the two dimensions.
• The width is the lesser dimension.
• Together they define the rectangular shape of the field.

Understanding dimensions is crucial when you want to calculate the field’s diagonal, as both measurements directly affect the solution of the problem.

Diagonal Calculation

The diagonal of a rectangle is the line segment that stretches from one corner of the rectangle to the opposite corner. It forms the hypotenuse of a right triangle whose other two sides are the width and the length of the rectangle. The Pythagorean Theorem is used to find the length of this diagonal. This theorem applies because a rectangle itself can be divided into two right triangles by its diagonal.
Here's how to calculate the diagonal:

• The formula is: \( d = \sqrt{L^2 + W^2} \), where \( L \) is the length and \( W \) is the width.
• First, square the length and the width.
• Add the squared values together.
• Take the square root of the resulting sum.

This process gives you the diagonal's length. For soccer fields, calculating the diagonal is important because it measures the maximum distance across the field, which can have various applications, such as determining the suitability for certain international competitions.