What is the area of a rectangle if x^2+y^2=625 and (x+y)^3-(x-y)^3=60690? x and y are the dimensions of rectangle.

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raziahmad's profile pic

raziahmad | (Level 1) eNoter

Posted on

Here I have tried to make the steps involved lesser than made by giorgiana1976 & I have copied her same steps as mine and further on I have simplified. So I seek permission of her & continue.

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The area of a rectangle is the product of the lengths of the sides of the rectangle:

A = x*y

Now, we'll work a bit over the second constraint from enunciation:

(x+y)^3-(x-y)^3=60690

As we can see, it is a difference of cubes:

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

(x+y)^3-(x-y)^3 = (x+y-x+y)[(x+y)^2 + (x+y)(x-y) + (x-y)^2]

We'll expand the squares inside brackets:

(x+y)^3-(x-y)^3 = 2y(x^2 + 2xy + y^2 + x^2 - y^2 + x^2 - 2xy + y^2)

We'll eliminate and combine like terms:

(x+y)^3-(x-y)^3 = 2y(x^2 + x^2 + x^2 + y^2)

But x^2 + y^2 = 625

(x+y)^3-(x-y)^3 = 2y(2x^2 + 625)

2y(2x^2 + 625) = 60690

y(2x^2 + 625) = 30345  .................................. (A)

These were steps made by giorgiana1976. thanks,

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Now,

from x^2 + y^2 = 625,

we have, x^2 = 625 - y^2, ...................................... (B)

Putting this value of x^2 in equation (A),

y(2(625-y^2) + 625) = 30345,

y(1250-2y^2 + 625) = 30345,

1875y -2y^3 = 30345,

or, 2y^3 - 1875y + 30345 = 0 -------------------- (C)

Value of 'y' can be found by solving above cubic equation.

And then value of 'x' fron equation (B).

And Hence  area of rectangle = [Value of 'x']*[Value of 'y']m

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The area of a rectangle is the product of the lengths of the sides of the rectangle:

A = x*y

Now, we'll work a bit over the second constraint from enunciation:

(x+y)^3-(x-y)^3=60690

As we can see, it is a difference of cubes:

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

(x+y)^3-(x-y)^3 = (x+y-x+y)[(x+y)^2 + (x+y)(x-y) + (x-y)^2]

We'll expand the squares inside brackets:

(x+y)^3-(x-y)^3 = 2y(x^2 + 2xy + y^2 + x^2 - y^2 + x^2 - 2xy + y^2)

We'll eliminate and combine like terms:

(x+y)^3-(x-y)^3 = 2y(x^2 + x^2 + x^2 + y^2)

But x^2 + y^2 = 625

(x+y)^3-(x-y)^3 = 2y(2x^2 + 625)

2y(2x^2 + 625) = 60690

y(2x^2 + 625) = 30345

We'll divide by (2x^2 + 625):

y = 30345/(2x^2 + 625)

Now, we'll substitute in the first constraint:

x^2 + [30345/(2x^2 + 625)]^2 = 625

x^2(4x^4 + 2500x^2 + 203125) = 625(4x^4 + 2500x^2 + 203125) - 920819025

We'll substitute x^2 by t:

t(4t^2 + 2500t + 203125) = 625*4t^2 + 1562500t -793865900

We'll remove the brackets:

4t^3 + 2500t^2 - 2500t^2 + 203125t -  1562500t + 793865900

We'll eliminate like terms:

4t^3 - 1359375t + + 793865900 = 0

we'll determine t and then x^2.

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