Problem 55 - 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

The diagonal for the minimum dimensions is approximately 118.74 meters. The diagonal for the maximum dimensions is approximately 133.15 meters.

Step 1: Identify the Problem

We need to find the diagonal length of a rectangle; thus, we can use the Pythagorean Theorem since the diagonal divides the rectangle into two right triangles. The diagonal is calculated as \( c = \sqrt{a^2 + b^2} \), where \( a \) is the length and \( b \) is the width.

Step 1: Calculate Minimum Diagonal

First, find the diagonal for the field at minimum dimensions:- Length \( a = 100 \) meters- Width \( b = 64 \) metersApply the formula for the diagonal:\[ c_{min} = \sqrt{100^2 + 64^2} = \sqrt{10000 + 4096} = \sqrt{14096} \approx 118.74 \text{ meters} \]

Step 2: Calculate Maximum Diagonal

Now, find the diagonal for the field at maximum dimensions:- Length \( a = 110 \) meters- Width \( b = 75 \) metersApply the same formula for the diagonal:\[ c_{max} = \sqrt{110^2 + 75^2} = \sqrt{12100 + 5625} = \sqrt{17725} \approx 133.15 \text{ meters} \]

Key Concepts

Diagonal of a Rectangle

The diagonal of a rectangle is a line segment that connects two opposite corners of a rectangle. This line effectively splits the rectangle into two congruent right triangles. Given the structure of a rectangle, calculating the length of this diagonal is an application of the Pythagorean Theorem.

The Pythagorean Theorem is a fundamental principle in geometry which states that in a right triangle, the square of the hypotenuse (which, in this context, is the diagonal) is equal to the sum of the squares of the two other sides. Therefore, if you know the length of the rectangle (7a7) and the width (7b7), the length of the diagonal (7c7) is given by the formula:

• \[ c = \sqrt{a^2 + b^2} \]

For example, in a rectangular soccer field, the diagonal can be a useful measure of the longest straight line across the field. It's particularly significant when considering dimensions such as for international matches, where specific ranges are used.

Rectangular Field Dimensions

Understanding the concepts of field dimensions is crucial, especially for standards like international soccer fields. Soccer fields have defined length and width specifications within which they should comply. In the context of your exercise, the rectangular soccer field has a length range of 100 to 110 meters and a width range of 64 to 75 meters.

These ranges mean that while the length and width can vary independently within these specifications, every potential configuration still forms a rectangle. The minimum and maximum dimensions should be calculated separately to determine the respective smallest and largest possible field sizes. This concept of ranges allows teams and venues to make specific adjustments to the field size while still staying within the accepted guidelines.

When determining configurations:

• The minimum field dimensions are 100 meters (length) by 64 meters (width).
• The maximum field dimensions are 110 meters (length) by 75 meters (width).

These configurations are useful for calculating the diagonal for both extremities.

Minimum and Maximum Calculations

To derive the diagonal length, both at the minimal and maximal dimensions of the field, we apply the formula for the hypotenuse of a right triangle using the respective dimensions.

For the minimum field dimensions of 100 meters in length and 64 meters in width, applying the Pythagorean Theorem yields:

• \[ c_{min} = \sqrt{100^2 + 64^2} = \sqrt{10000 + 4096} = \sqrt{14096} \]

This results in a diagonal length of approximately 118.74 meters.

For the maximum field dimensions of 110 meters in length and 75 meters in width, the calculations are:

• \[ c_{max} = \sqrt{110^2 + 75^2} = \sqrt{12100 + 5625} = \sqrt{17725} \]

The length of this diagonal comes out to be approximately 133.15 meters.

These calculations highlight the importance of analyzing both the smallest and largest setups allowable to prepare for real-world applications, such as setting up a field for professional events.