Answer :
To solve the problem of determining the largest rectangular area Jessica can create with 900 feet of fencing, we need to find the optimal dimensions for the rectangle.
### Step 1: Define Variables
Let:
- [tex]\( x \)[/tex] be the width of the rectangle (in feet).
- [tex]\( 450 - x \)[/tex] be the length of the rectangle (in feet), since she uses 900 feet of fencing, with half for each pair of opposite sides.
### Step 2: Express Area as a Function
The area [tex]\( A \)[/tex] of a rectangle is given by the product of its width and length:
[tex]\[ A = x \cdot (450 - x) \][/tex]
### Step 3: Simplify the Area Function
[tex]\[ A = 450x - x^2 \][/tex]
### Step 4: Find the Maximum Area
To find the value of [tex]\( x \)[/tex] that maximizes the area, take the derivative of the area function with respect to [tex]\( x \)[/tex] and set it to zero:
[tex]\[ \frac{dA}{dx} = 450 - 2x \][/tex]
Set the derivative to zero to find critical points:
[tex]\[ 450 - 2x = 0 \][/tex]
[tex]\[ 2x = 450 \][/tex]
[tex]\[ x = 225 \][/tex]
### Step 5: Calculate the Corresponding Length
If [tex]\( x = 225 \)[/tex] feet, then:
[tex]\[ 450 - x = 450 - 225 = 225 \text{ feet} \][/tex]
### Step 6: Calculate the Maximum Area
Substitute [tex]\( x = 225 \)[/tex] feet back into the area function:
[tex]\[ A = 225 \cdot 225 = 50625 \text{ square feet} \][/tex]
Therefore, the largest rectangular area Jessica can create is:
[tex]\[ 50,625 \text{ square feet} \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{50,625 \text{ square feet} \][/tex]
### Step 1: Define Variables
Let:
- [tex]\( x \)[/tex] be the width of the rectangle (in feet).
- [tex]\( 450 - x \)[/tex] be the length of the rectangle (in feet), since she uses 900 feet of fencing, with half for each pair of opposite sides.
### Step 2: Express Area as a Function
The area [tex]\( A \)[/tex] of a rectangle is given by the product of its width and length:
[tex]\[ A = x \cdot (450 - x) \][/tex]
### Step 3: Simplify the Area Function
[tex]\[ A = 450x - x^2 \][/tex]
### Step 4: Find the Maximum Area
To find the value of [tex]\( x \)[/tex] that maximizes the area, take the derivative of the area function with respect to [tex]\( x \)[/tex] and set it to zero:
[tex]\[ \frac{dA}{dx} = 450 - 2x \][/tex]
Set the derivative to zero to find critical points:
[tex]\[ 450 - 2x = 0 \][/tex]
[tex]\[ 2x = 450 \][/tex]
[tex]\[ x = 225 \][/tex]
### Step 5: Calculate the Corresponding Length
If [tex]\( x = 225 \)[/tex] feet, then:
[tex]\[ 450 - x = 450 - 225 = 225 \text{ feet} \][/tex]
### Step 6: Calculate the Maximum Area
Substitute [tex]\( x = 225 \)[/tex] feet back into the area function:
[tex]\[ A = 225 \cdot 225 = 50625 \text{ square feet} \][/tex]
Therefore, the largest rectangular area Jessica can create is:
[tex]\[ 50,625 \text{ square feet} \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{50,625 \text{ square feet} \][/tex]
