Problem 25 - Prealgebra

Question

Set up the following problems the same way you set up Problems 1-22. Then use a calculator to do the calculations. The rules for soccer state that the playing field must be from 100 to 120 yards long and 55 to 75 yards wide. The 1999 Women's world Cup was played at the Rose Bowl on a playing field 116 yards long and 72 yards wide. The diagram below shows the smallest possible soccer field, the largest possible soccer field, and the soccer field at the Rose Bowl. (picture not copy) a. Percent Increase A team plays on the smallest field, then plays in the Rose Bowl. What is the percent increase in the area of the playing field from the smallest field to the Rose Bowl? Round to the nearest tenth of a percent. b. Percent increase A team plays a soccer game in the Rose Bowl. The next game is on a field with the largest dimensions. What is the percent increase in the area of the playing field from the Rose Bowl to the largest field? Round to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The percent area increase from the smallest field to the Rose Bowl is approximately 51.8%, and from the Rose Bowl to the largest field is 7.8%.

Step 1: Calculate Smallest Field Area

The smallest possible soccer field measures 100 yards in length and 55 yards in width. To find its area, multiply these two dimensions: \(100 \times 55 = 5500\) square yards.

Step 2: Calculate Rose Bowl Field Area

The Rose Bowl's field is 116 yards long and 72 yards wide. Thus, its area is \(116 \times 72 = 8352\) square yards.

Step 3: Calculate Percent Increase to Rose Bowl Field

To find the percent increase in the area going from the smallest field to the Rose Bowl, use the formula: \(\text{Percent Increase} = \frac{\text{New Area} - \text{Old Area}}{\text{Old Area}} \times 100\). Substituting the values: \(\frac{8352 - 5500}{5500} \times 100 \approx 51.8\%\).

Step 4: Calculate Largest Field Area

The largest possible soccer field measures 120 yards in length and 75 yards in width. Therefore, its area is \(120 \times 75 = 9000\) square yards.

Step 5: Calculate Percent Increase to Largest Field

To find the percent increase from the Rose Bowl field to the largest field, use the same percent increase formula: \(\frac{9000 - 8352}{8352} \times 100 \approx 7.8\%\).

Key Concepts

Understanding Area Calculations

Area calculations are essential for determining the size of a space, such as a soccer field. To calculate the area of a rectangle, you multiply its length by its width. This gives you the total number of square units that fit into the rectangle.
In the context of soccer fields:

• The smallest soccer field measures 100 yards by 55 yards. Its area is calculated as \(100 \times 55 = 5500\) square yards.
• The Rose Bowl field, slightly larger, measures 116 yards by 72 yards, resulting in an area of \(116 \times 72 = 8352\) square yards.
• The largest possible field, at 120 yards by 75 yards, has an area of \(120 \times 75 = 9000\) square yards.

These calculations show how the dimensions of a field directly influence its total area, which is critical for understanding spatial layouts when designing or analyzing field parameters.

Soccer Field Dimensions

Soccer field dimensions can vary, but there are regulations that the fields must adhere to. According to the rules:

• The length of a soccer field must be between 100 and 120 yards.
• The width must be between 55 and 75 yards.

These measurements allow flexibility in field size, accommodating different stadiums and venues. Knowing the exact dimensions is important for both strategy and logistics in soccer matches. The Rose Bowl, used for the 1999 Women's World Cup, exemplifies a larger field within these limits, measuring 116 yards in length and 72 yards in width. This variation in field size can affect gameplay, as wider and longer fields may require different tactics and greater stamina from players.

Step-by-Step Problem-Solving Strategies

Approaching problems like percent increase calculations involves clear, systematic steps:

• First, calculate the area of each field involved in the comparison. This provides the necessary baseline data.
• Next, use the percent increase formula: \[\text{Percent Increase} = \frac{\text{New Area} - \text{Old Area}}{\text{Old Area}} \times 100\]This formula helps determine how much larger one area is compared to another.
• Apply the formula to your specific areas—such as from the smallest field to the Rose Bowl or from the Rose Bowl to the largest field.

By following these steps, you ensure consistent and accurate results that are easy to verify. It also builds confidence in handling similar future scenarios. These approaches are valuable not just in calculating field dimensions, but in tackling diverse mathematical problems.