Question

The infield of a baseball diamond is in the shape of a rhombus. An infield cover with dimensions of 85 feet by 100 feet is used to protect the field during rainy weather. Will the cover protect the entire infield? Explain.
90 feet
90 feet
90 feet
90 feet

Answer 1

To determine if the cover will protect the entire infield (which is shaped like a rhombus), we need to check if the area of the cover is equal to or greater than the area of the rhombus.

1. **Calculate the area of the cover**:
The area of the rectangle (cover) can be calculated using the formula:
\[
\text{Area}_{\text{cover}} = \text{Length} \times \text{Width} = 85 \, \text{feet} \times 100 \, \text{feet} = 8500 \, \text{square feet}
\]

2. **Determine the dimensions of the rhombus infield**:
A baseball diamond's infield is typically a square rotated 45 degrees (forming a rhombus), and each side is about 90 feet (as indicated in your data). The diagonals of the rhombus can be calculated with the Pythagorean theorem since it is a 45-45-90 triangle. The diagonals \(d_1\) and \(d_2\) can be found as follows:
\[
d_1 = d_2 = 90 \, \text{feet} \times \sqrt{2} = 90\sqrt{2} \approx 127.28 \, \text{feet}
\]

3. **Calculate the area of the rhombus**:
The area of a rhombus can also be calculated using the lengths of its diagonals:
\[
\text{Area}_{\text{rhombus}} = \frac{1}{2} \times d_1 \times d_2
\]
Here \(d_1\) and \(d_2\) are both approximately \(127.28\) feet. Therefore, we have:
\[
\text{Area}_{\text{rhombus}} = \frac{1}{2} \times 127.28 \, \text{feet} \times 127.28 \, \text{feet} \approx \frac{1}{2} \times 16200 \, \text{square feet} \approx 8100 \, \text{square feet}
\]

4. **Comparison**:
Area of the cover: \(8500\) square feet
Area of the rhombus infield: \(8100\) square feet

Since the area of the cover (8500 square feet) is greater than the area of the rhombus infield (8100 square feet), the cover will indeed protect the entire infield.