Alright, let's walk through the problem step-by-step to find the area of the field excluding the pond and the cost of planting vegetables.
### Step 1: Calculate the Radius of the Pond
The diameter of the pond is given as 28 feet. The radius [tex]\( r \)[/tex] is half the diameter:
[tex]\[ r = \frac{28}{2} = 14 \text{ feet} \][/tex]
### Step 2: Calculate the Area of the Pond
The area [tex]\( A \)[/tex] of a circle is calculated using the formula:
[tex]\[ A_{\text{pond}} = \pi \times r^2 \][/tex]
Substituting the radius:
[tex]\[ A_{\text{pond}} \approx 3.14159 \times 14^2 \approx 615.752 \text{ square feet} \][/tex]
### Step 3: Calculate the Area of the Field
The field is rectangular with a length of 40 feet and a breadth of 40 feet. The area [tex]\( A \)[/tex] of the rectangle is:
[tex]\[ A_{\text{field}} = \text{length} \times \text{breadth} \][/tex]
[tex]\[ A_{\text{field}} = 40 \times 40 = 1600 \text{ square feet} \][/tex]
### Step 4: Calculate the Area of the Field Excluding the Pond
To find the area of the field excluding the pond, subtract the area of the pond from the area of the field:
[tex]\[ A_{\text{excluding pond}} = A_{\text{field}} - A_{\text{pond}} \][/tex]
[tex]\[ A_{\text{excluding pond}} = 1600 - 615.752 = 984.248 \text{ square feet} \][/tex]
### Step 5: Calculate the Cost of Planting Vegetables
The cost of planting vegetables is given as [tex]$4 per square foot. To find the total cost, multiply the area of the field excluding the pond by the cost per square foot:
\[ \text{Cost} = A_{\text{excluding pond}} \times \text{cost per square foot} \]
\[ \text{Cost} = 984.248 \times 4 \approx 3937 \text{ dollars} \]
Thus, the area of the field excluding the pond is approximately 984.248 square feet, and the cost of planting vegetables in the remaining area would be about $[/tex]3937.
