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)= (type an expression using x as the variable )

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)= (type an expression using x as the variable )

Answer

Explanation:

Step1: Express the length parallel to shoreline

Let ( x ) be the length of a side perpendicular to the shoreline. The total length of rope is ( 1200 ) yd. Since there are two sides perpendicular to the shoreline (( 2x )) and one side parallel to the shoreline (( y )), we have ( 2x + y=1200 ), so ( y = 1200 - 2x ).

Step2: Write the area function

The area of a rectangle ( A=xy ). Substitute ( y = 1200 - 2x ) into the area formula. Then ( A(x)=x(1200 - 2x)=1200x-2x^{2} ).

Answer:

( A(x)=1200x - 2x^{2} )