Answer to Question #264696 in Geometry for Ali

Question #264696

The area of a rectangle is: x²-1/2


a)What must be the length of this rectangle if its width is x-1



b)For what value of x is this rectangle a square?


1
Expert's answer
2021-11-16T14:09:44-0500

We know that Area of a rectangle is

Area=(lengthbreadth)unit²Area=(length*breadth)unit²

A=lbA=l*b

Where A=x²½A=x²-½

(a) we have width, bb to be:

b=x1b=x-1

Let the length be yy

A=lbA=l*b

y(x1)=x²½y(x-1)=x²-½

Dividing both sides by x1x-1

y=x²½x1y=\frac{x²-½}{x-1}

y=2x²12x1y=\frac{2x²-1}{2x-1} unit

Which gives the expression for the length of the rectangle as yy inform of xx.


(b) We have the formula for Area of a square to be:

A=l²A=l² (in unit squared)

Where l=x1l=x-1

l²=Al²=A

(x1)²=x²½(x-1)²=x²-½

x²2x+1=x²½x²-2x+1=x²-½

By collecting the like terms

x²x²2x=½1x²-x²-2x=-½-1

2x=32-2x=-\frac{3}{2}

Multiply through by -1

2x=322x=\frac{3}{2}

Cross multiply

4x=34x=3

x=34x=\frac{3}{4}

Therefore, the value of xx for which the rectangle is a square is 34\frac{3}{4}


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