Problem 23 - Prealgebra

Question

Find the perimeter and area of the soccer field described at the left. Freddy Adu became the youngest professional player in modern American team sports history when he joined D.C. United at 14 years of age. Soccer is played on a rectangular field that is usually 120 yards long and 75 yards wide.

Step-by-Step Solution

Answer

The perimeter is 390 yards and the area is 9000 square yards.

Step 1: Identify Dimensions

Read the problem and note the dimensions for the soccer field: 120 yards long and 75 yards wide. These will be used to calculate both the perimeter and the area of the field.

Step 2: Formula for Perimeter of Rectangle

Recall the formula for the perimeter of a rectangle: \[P = 2 imes ( ext{length} + ext{width})\] where the length is 120 yards and the width is 75 yards.

Step 3: Calculate Perimeter

Substitute the given dimensions into the perimeter formula: \[P = 2 imes (120 + 75) = 2 imes 195 = 390 ext{ yards}\]So, the perimeter of the soccer field is 390 yards.

Step 4: Formula for Area of Rectangle

Recall the formula for the area of a rectangle: \[A = ext{length} imes ext{width}\] where the length is 120 yards and the width is 75 yards.

Step 5: Calculate Area

Substitute the given dimensions into the area formula: \[A = 120 imes 75 = 9000 ext{ square yards}\]So, the area of the soccer field is 9000 square yards.

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It is all about understanding how different figures are formed and measured. When solving problems related to perimeter and area, we use geometric principles to determine the size and boundary of figures such as rectangles, squares, circles, and more.

Perimeter refers to the total distance around the outside of a shape. For any polygon, such as a rectangle, you find the perimeter by summing the lengths of all its sides. For regular shapes like rectangles, it is often more efficient to apply specific formulas. In contrast, the area measures the size of the surface of a shape and is expressed in square units.

• Formulas:
  • Perimeter of rectangle: \( P = 2 \times (\text{length} + \text{width}) \)
  • Area of rectangle: \( A = \text{length} \times \text{width} \)
• Importance: Understanding these concepts can help in real-life applications such as construction, sports, and design.

Rectangular Field

To understand fields and their use in sports, it's essential to know how they are shaped and measured. A soccer field is a common example of a rectangular field, which is a shape with four sides, where opposite sides are equal length. This shape allows for efficient use of space, providing a large area using minimal boundary length.

Knowing the dimensions of such fields helps in various calculations, such as determining where to place lines and goals. These calculations can also impact the design and layout of stadiums and sports facilities.

• Attributes:
  • Rectangles have opposite sides that are equal.
  • All angles in a rectangle are right angles, measuring 90 degrees each.
• Uses: Rectangular fields are used in many sporting activities including soccer, football, and rugby.

Basic Calculations

When dealing with practical problems, basic calculations are significant as they form the foundation of more complex operations. Calculating the perimeter and area of a rectangular field requires some simple steps that help ensure accurate results.

In this context, you'll use basic arithmetic operations to reach a solution. Recognize the given dimensions, apply the correct formulas, and substitute the numbers to determine the perimeter and area. 

• Steps to Calculate:
  • Identify dimensions: length and width of the field.
  • Apply the formulas for perimeter and area of rectangles.
  • Substitute the values and solve.
• Example: For a soccer field with a length of 120 yards and a width of 75 yards, the perimeter is calculated as \( 2 \times (120 + 75) = 390 \) yards, while the area is \( 120 \times 75 = 9000 \) square yards.