The point P lies in the first quadrant on the graph of the line y= 7-3x. From the point P, perpendiculars are drawn to both the x-axis and y-axis. What is the largest possible area for the rectangle thus formed?

1 Answer
Apr 9, 2017

49/12" sq.unit."4912 sq.unit.

Explanation:

Let M and NMandN be the feet of bot from P(x,y)P(x,y) to the X-X Axis

and Y-Y Axis, resp., where,

P in l={(x,y) | y=7-3x, x>0; y>0} sub RR^2....(ast)

If O(0,0) is the Origin, the, we have, M(x,0), and, N(0,y).

Hence, the Area A of the Rectangle OMPN, is, given by,

A=OM*PM=xy," and, using "(ast), A=x(7-3x).

Thus, A is a fun. of x, so let us write,

A(x)=x(7-3x)=7x-3x^2.

For A_(max), (i) A'(x)=0, and, (ii) A''(x)<0.

A'(x)=0 rArr 7-6x=0 rArr x=7/6, >0.

Also, A''(x)=-6," which is already "<0.

Accordingly, A_(max)=A(7/6)=7/6{7-3(7/6)}=49/12.

Therefore, the largest possible area of the rectangle is 49/12" sq.unit."

Enjoy Maths.!