a lifeguard needs to rope off a rectangular swimming area in front of long lake beach, using 1200 yd of rope…

a lifeguard needs to rope off a rectangular swimming area in front of long lake beach, using 1200 yd of rope and floats. what dimensions of the rectangle will maximize the area? what is the maximum area? (note that the shoreline is one side of the rectangle.) let x be the length of a side of the rectangle perpendicular to the shoreline. write the objective function for the area in terms of x a(x) = 1200x - 2x² (type an expression using x as the variable.) the length of the shorter side of the rectangular region is 300 yd the length of the longer side of the rectangular region is 600 yd the maximum area of the rectangular region is

a lifeguard needs to rope off a rectangular swimming area in front of long lake beach, using 1200 yd of rope and floats. what dimensions of the rectangle will maximize the area? what is the maximum area? (note that the shoreline is one side of the rectangle.) let x be the length of a side of the rectangle perpendicular to the shoreline. write the objective function for the area in terms of x a(x) = 1200x - 2x² (type an expression using x as the variable.) the length of the shorter side of the rectangular region is 300 yd the length of the longer side of the rectangular region is 600 yd the maximum area of the rectangular region is

Answer

Explanation:

Step1: Recall the formula for the area of a rectangle

The area of a rectangle is (A = length\times width). We are given (A(x)=1200x - 2x^{2}), which is a quadratic function of the form (y = ax^{2}+bx + c), where (a=- 2), (b = 1200), and (c = 0).

Step2: Use the formula for the vertex of a quadratic function

For a quadratic function (y = ax^{2}+bx + c), the (x) - coordinate of the vertex (which gives the maximum value when (a<0)) is (x=-\frac{b}{2a}). Substituting (a=-2) and (b = 1200) into (x =-\frac{b}{2a}), we get (x=-\frac{1200}{2\times(-2)}=300).

Step3: Find the maximum area

Substitute (x = 300) into the area function (A(x)=1200x-2x^{2}). [ \begin{align*} A(300)&=1200\times300-2\times(300)^{2}\ &=360000-2\times90000\ &=360000 - 180000\ &=180000 \end{align*} ]

Answer:

(180000) square yards