Final answer:
Using the Pythagorean theorem, the length of the rectangular playing field is calculated to be approximately 90.51 feet, given the diagonal measures 96 feet and the width measures 32 feet.
Step-by-step explanation:
The student is asking for help to determine the length of a rectangular playing field, given its diagonal and width measurements. To find the length of the field, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this scenario, the diagonal of the rectangle acts as the hypotenuse, the width is one of the sides, and the length we need to find is the other side. Assuming the measurements are correct and the field is a perfect rectangle, the formula is expressed as c^2 = a^2 + b^2, where c is the diagonal (96 feet), a is the width (32 feet), and b is the unknown length.
First, we square the width: 32 feet * 32 feet = 1024 square feet. Next, we square the diagonal measurement: 96 feet * 96 feet = 9216 square feet. Now we subtract the square of the width from the square of the diagonal to find the square of the length: 9216 - 1024 = 8192 square feet. To find the length, we take the square root of 8192, which gives us a length of 90.51 feet, when rounded to two decimal places. Therefore, the length of the field is approximately 90.51 feet.