To solve the problem of maximizing the area for a rectangular field bounded on one side by a lake and fenced on the other three sides with 800 meters of fencing, let's follow these steps:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the length of the field parallel to the lake.
- Let [tex]\( y \)[/tex] be the width of the field perpendicular to the lake.
2. Establish the Constraint:
- Since the field is bounded on one side by the lake, the fencing is used for the other three sides: two widths (perpendicular sides) and one length (parallel side).
- Therefore, the total length of fencing used is given by the equation:
[tex]\[
x + 2y = 800
\][/tex]
- From this equation, we can express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 800 - 2y
\][/tex]
3. Area of the Field:
- The area [tex]\( A \)[/tex] of the rectangular field can be expressed as:
[tex]\[
A = x \times y
\][/tex]
- Substitute the expression for [tex]\( x \)[/tex] from the constraint equation:
[tex]\[
A = (800 - 2y) \times y
\][/tex]
4. Form the Area Function:
- This gives us a quadratic equation in terms of [tex]\( y \)[/tex]:
[tex]\[
A = 800y - 2y^2
\][/tex]
5. Differentiate to Find the Maximum Area:
- To find the value of [tex]\( y \)[/tex] that maximizes the area, we take the derivative of [tex]\( A \)[/tex] with respect to [tex]\( y \)[/tex] and set it equal to zero:
[tex]\[
\frac{dA}{dy} = 800 - 4y = 0
\][/tex]
- Solving for [tex]\( y \)[/tex]:
[tex]\[
800 - 4y = 0 \implies 4y = 800 \implies y = 200
\][/tex]
6. Find the Corresponding [tex]\( x \)[/tex] Value:
- Substitute [tex]\( y = 200 \)[/tex] back into the equation [tex]\( x = 800 - 2y \)[/tex]:
[tex]\[
x = 800 - 2 \times 200 = 800 - 400 = 400
\][/tex]
7. Calculate the Maximum Area:
- The maximum area [tex]\( A \)[/tex] can now be calculated using:
[tex]\[
A = x \times y = 400 \times 200 = 80000
\][/tex]
8. Conclusion:
- The dimensions of the rectangular field that will produce a maximum area are:
[tex]\[
\text{Width} (y) = 200 \text{ meters}, \quad \text{Length} (x) = 400 \text{ meters}
\][/tex]
- The maximum area of the field is [tex]\( 80000 \)[/tex] square meters.
