Question 33, 1.10.21 Score: \( 64.48 \%, 67.7 \) of You have 950 feet of fencing to enclose a rectangular field. Express the area of the field, A , as a function of one of its dimensions, x . \( \mathrm{A}(\mathrm{x})=\square \)

To express the area \( A \) of a rectangular field as a function of one of its dimensions \( x \), we first need to define the other dimension. Since you have 950 feet of fencing, the perimeter \( P \) is given by the equation \( P = 2x + 2y = 950 \). From this, we can solve for \( y \): \[ y = \frac{950 - 2x}{2} = 475 - x. \] Then, the area \( A \) can be expressed as: \[ A(x) = x \cdot y = x \cdot (475 - x) = 475x - x^2. \] Now you have \( A(x) = 475x - x^2 \), which represents the area of the field as a function of dimension \( x \). To maximize your field area, remember that it forms a parabola when plotted from the function \( A(x) = 475x - x^2 \), which opens downward. The maximum area occurs at the vertex, and since the coefficient of \( x^2 \) is negative, this ensures it is a maximum point! You can find this by using the vertex formula \( x = -\frac{b}{2a} \) if you need to find the specific dimensions for optimal area.