Question: Jordan wants to enclose an outdoor rectangular garden where one side of the enclosure is bounded by their house. If they have 118 feet of fencing to work with for the remaining three sides of the garden, find the dimensions of the rectangle that will result in a maximum area and find the maximum area.
Side length opposite the house:
?
Side length perpendicular to the house:
?
Maximum area:
?
1. **State the problem:** Jordan wants to build a rectangular garden with one side along the house, so fencing is needed for only three sides. The total fencing available is $118$ feet. We need to find the dimensions that maximize the area and the maximum area itself.
2. **Define variables:**
Let $x$ be the length of the side opposite the house (the side parallel to the house).
Let $y$ be the length of each of the two sides perpendicular to the house.
3. **Write the fencing constraint:**
Since the house forms one side, fencing is needed for the opposite side and the two perpendicular sides:
$$x + 2y = 118$$
4. **Express $x$ in terms of $y$:**
$$x = 118 - 2y$$
5. **Write the area function:**
The area $A$ of the rectangle is:
$$A = x \times y$$
Substitute $x$:
$$A = (118 - 2y) y = 118y - 2y^2$$
6. **Maximize the area:**
This is a quadratic function in $y$:
$$A(y) = -2y^2 + 118y$$
The maximum occurs at the vertex of the parabola.
7. **Find the vertex:**
The vertex $y$-coordinate is given by:
$$y = -\frac{b}{2a} = -\frac{118}{2 \times (-2)} = \frac{118}{4} = 29.5$$
8. **Find $x$ using $y=29.5$:**
$$x = 118 - 2(29.5) = 118 - 59 = 59$$
9. **Calculate the maximum area:**
$$A = x \times y = 59 \times 29.5 = 1740.5$$
**Final answers:**
- Side length opposite the house: $59$ feet
- Side length perpendicular to the house: $29.5$ feet
- Maximum area: $1740.5$ square feet