A rectangle is inscribed in the region bounded by the x-axis, the y-axis, and the graph of x+2y-8=0 as shown in the figure.

Question

Numerical, Graphical, and Analytical Analysis - A

rectangle is inscribed in the region bounded by the x-axis,

the y-axis, and the graph of x+2y-8=0 as shown in

the figure.

A) Write the area A of the rectangle as a function of x.

Determine the domain of the function in the context of

the problem.

D) Write the area function in standard form to find

algebraically the dimensions that will produce a

maximum area.

Answer 1

Derivatives are not a standard part of a precalculus course. For quadratics, x=-b/2/a is shown to be the x value associated with the maximum/minimum point of the function.

So, -4/2/(-0.5) = 4 = maximum x value point, the same as with the derivative.

Answer 2

area of rectangle in quadrant 1, inscribed between three lines

x+2y-8=0

or

y=-x/2+4

x=0, and y=0

domain is (0,8)

Area = xy =x(-x/2 +4)

A(x)= 4x-.5x^2

A(x)=-.5x^2 +4x

A'(x)=-x+4=0

x=4, y=2

range is (0,8]

max Area=xy=4(2)= 8

A= -.5x2 +4x in standard form

A=-.5(x^2 +8x+16)+8

A=-.5(x-4)^2 +8 in vertex form with 8=max A when x= 4

graphically,the max area is found by using the midpoint of the line segment

connecting the two intercepts of the given line x+2y -8=0

y intercept= (0,4), x intercept =(8,0)

midpoint =(0+8/2, 4/2+0) =(4,2)

4x2 =8 =max area