Question: A carpenter is building a rectangular shed with a fixed perimeter of 50 feet. Find the dimensions of the largest floor area of the shed that can be built and the maximum floor area of the shed.
Side length:
Maximum area:
1. **State the problem:** We need to find the dimensions of a rectangular shed with a fixed perimeter of $50$ feet that maximize the floor area.
2. **Define variables:** Let the length be $x$ feet and the width be $y$ feet.
3. **Write the perimeter formula:** The perimeter $P$ of a rectangle is given by $$P = 2x + 2y.$$ Given $P = 50$, we have:
$$2x + 2y = 50.$$
4. **Express one variable in terms of the other:**
$$2y = 50 - 2x$$
$$y = \frac{50 - 2x}{2} = 25 - x.$$
5. **Write the area formula:** The area $A$ of the rectangle is:
$$A = xy.$$ Substitute $y$:
$$A = x(25 - x) = 25x - x^2.$$
6. **Maximize the area:** This is a quadratic function $A(x) = -x^2 + 25x$ which opens downward, so its maximum is at the vertex.
7. **Find the vertex:** The $x$-coordinate of the vertex is given by:
$$x = -\frac{b}{2a} = -\frac{25}{2(-1)} = \frac{25}{2} = 12.5.$$
8. **Find the corresponding $y$:**
$$y = 25 - x = 25 - 12.5 = 12.5.$$
9. **Calculate the maximum area:**
$$A = 12.5 \times 12.5 = 156.25.$$
**Answer:**
- Side lengths: $12.5$ feet by $12.5$ feet (a square)
- Maximum floor area: $156.25$ square feet