Writing the area of a rectangle as a function of an x-coordinate

Apr 2010
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9
This is the question:

Express the area of the rectangle shown in the accompanying figure as a function of the x-coordinate of the point P.



My attempt:

I know the area of a rectangle is A = xy

Since it needs to be a function of the x-coordinate, I need to find y in terms of x, so I can substitute it and have only 'x' as a variable. But I'm not sure how to calculate y.

Any assistance would be great.
 
Jun 2007
1,252
683
Medgidia, Romania
Let \(\displaystyle A(a,0), \ B(0,b)\).

The equation of the line AB is:

\(\displaystyle \displaystyle\frac{x-a}{-a}=\displaystyle\frac{y}{b}\Rightarrow y=\frac{b}{a}(a-x)\)

The point P is on the line AB, so the coordinates of P satisfy the equation of the line AB.

Now, replace y in the area of the rectangle:

\(\displaystyle A=\displaystyle\frac{b}{a}x(a-x)\)
 

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Jan 2006
5,854
2,553
Germany
This is the question:

Express the area of the rectangle shown in the accompanying figure as a function of the x-coordinate of the point P.



My attempt:

I know the area of a rectangle is A = xy

Since it needs to be a function of the x-coordinate, I need to find y in terms of x, so I can substitute it and have only 'x' as a variable. But I'm not sure how to calculate y.

Any assistance would be great.
1. The point P is placed on a straight line which passes through the points R(0, b) and S(a, 0).

2. Determine the equation of the line RS which will provide you with the term of the veariable y.

3. For confirmation only: I've got \(\displaystyle y = -\frac ba x + b\)
 
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Dec 2009
1,506
434
Russia
Or:

ab/2 - {x(b-y)}/2 - {y(a-x)}/2=

= {ab - xb -xy -ya+yx}/2

= {ab - ay}/2 =AREA OF RECTANGLE = xy

==> ab - ay =xy

ab=ay+xy

==>
ab= y(a+x)

y=ab/{a+x}

so xy= xab/{a+x}
 
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