The combined area of a square and rectange.

Question

The combined area of a square and a rectangle is 229 square centimeters. The width of the rectangle is 3 centimeters more than the length of a side of the square, and the length of the rectangle is 4 centimeters more than its width. Find the dimensions of the square and the rectangle.

Answer 1

w x l + s² = 229      and rearranged    w x l = 229 - s²

 

w = s + 3    and   s = w - 3

 

l = w + 4

 

using substitution ,  replace l in first equation with w+4:

 

w x (w + 4) = 229 - s²       and rewrite s in terms of w:

 

w x (w + 4) = 229 - (w - 3)²  

 

 Now simplify with distributive property,use FOIL on the binomial squared and combine like terms:

 

       w² + 4w = 229 - (w -3)x(w-3)

 

         w² + 4w = 229 - (w² - 6w + 9)

 

Using inverse operations, isolate w terms to the left side and create a quadratic equation in standard form:

 

       2w² - 2w - 220 = 0    Factor out the 2:

 

2(w² - w - 110) = 0     Using the diamond method, we need to find two factors of -110 whose sum is -1; -11 times 10 is -110 and -11 + 10 is -1. Voila! Those factors are -11 and 10. 

 

Now write the quadratic as the product of two binomials using these factors:

        2 (w - 11) (w + 10) = 0 therefore the zeros are {11 and -10}.  -10 is an extraneous solution because you cannot have a negative difference. W is 11.

 

Now, go back to the given information at the top, and you will get the following, given that w = 11:

 

   length = 15 because   length = width + 4

 

         side of square = 8 because    side = w - 3

 

And, finally

 

11cm x 15cm + 8²cm = 165cm²+ 84cm²= 229cm²