What is the length of the side of the smaller square?

Question

The sum of the areas of two geometric squares is 1000. The side of one square is 10 less than 2/3 of the side of the other square. What is the length of the side of the smaller square?

Answer 1

 

Draw the two squares and label the side of the larger square x and the side of the smaller square ((2/3)x -10).  Note that we will solve for x then report ((2/3)x-10).

The sum of the areas is 1000, so let’s add the areas:

   Area of larger square       +    area of smaller square     =    1000

         x^2                          +   ((2/3)x-10)^2                       = 1000

        x^2                + (4/9)x^2 – (20)(2/3)x + 100             = 1000

              [multiply everything by 9 to get rid of fraction]

        9x^2              + 4x^2  -  120x              + 900             = 9000

      13x^2 – 120x = 8100

       13x^2 -120x -8100 = 0

       (x-30)(x+20.8) = 0           [use quadratic equation]

   Either x=30 or x=-20.8     but x can’t be negative

 

If x=30, then ((2/3)x-10)=10

The length of the side of the smaller square is 10

 

Checking:

  Is  30^2 + 10^2 = 1000   ?

        900 + 100 = 1000   ?

          1000 = 1000   ? yes, the total area is 1000

 

Answer 2

Dear Chaoyun,

 

 1.    First of all, it may be a good idea to draw the models of the two squares and label what you know for their sides.

 

2.  If the factoring of the quadratic is difficult, you may want to use the quadratic equation to find the solutions. 

 

3.  Finally, you may want to check your work to see if X=30 is your answer.  Let A_{1=larger area   and} A₂=smaller area.  The story problem states that the two areas together are 1000 sq. units.   Therefore, Area 1 + Area 2= 1000.  Substitute X=30 into each equation for a side, but don't forget to multiply the sides of that particular square to get the area of each before finding the sum.  A common mistake is not to do so.

 

If I have helped a little, please give me a thumbs.

 

 

 

 

 

 

 

                                                                                 Susan C.

Answer 3

Designate the side length of one of the squares as "x". We'll make the larger square's side "x". The side of the smaller square is 10 less than 2/3 of the side of the larger square. Therefore, the side of the smaller square is (2/3)x - 10.

 

The area of the larger square is x². The area of the smaller square is ((2/3)x - 10)², or (4/9)x² - (40/3)x + 100.

 

Add these two areas to get 1000. That yields a quadratic equation: x² + (4/9)x² - (40/3)x + 100 = 1000. Combine like terms to get (13/9)x² - (40/3)x + 100 = 1000. Solve the quadratic equation for "x". Here, "x" must be positive, since it's the side length of a square.

 

Once you have "x", you must still do one more thing! Remember that "x" was the length of the LARGER square's side. The problem, however, is asking for the length of the SMALLER square's side. From your "x", plug that value into (2/3)x - 10, to find the answer.

Answer 4

Let the side of one square be x, then the side of the other square is 2/3x-10

x²+( 2/3x-10)²=1000

x²+4/9x²+100-40/3x=1000

13/9x²-40/3x-900=0

13x²-120x-8100=0

(x-30)(13x+270)=0

since side length has to be positive,

x=30

2/3x-10 = 20-10 = 10

so the smaller square has length 10.