Word Problem

Question

A rectangular rug is placed on a rectangular floor. The width of the floor is 6 feet greater than the length, x, of the floor. The width of the rug is 4 feet less than the width of the floor. The length of the rug is 8 feet less than the width of the rug. Write a function, R(x), that represents the area of the floor not covered by the rug.

I need the steps, along with the answer for this problem please! 

Answer 1

Okay, lets start with the picture/drawing. A rug is being placed on a floor, so it is safe to assume that the floor is larger than the rug is.

 

Now, the "width of the floor is 6 feet greater than the length, x, of the floor" is another way of saying that the following:

 

Width of the floor: x+6

Length of the floor: x

 

Now, you can relate the rest of the problem to these values and figure out the area that would be occupied by the rug. The "width of the rug is 4 feet LESS than the width of the floor" means subtract 4 from our previous width equation."

 

Width of the rug: x+6-4 = x+2

 

"The length of the rug is 8 feet less than the width of the rug" means subtract 8 from the width of the rug equation:

 

Length of the rug: x+2-8 = x-6

 

So, you end up with the following equations:

 

Width of the floor: =x+6
Length of the floor:= x

Width of the rug:   = x+2

Length of the rug:  =x-6

 

Now, we know the area of rectangles is simply length*width. So, multiple the width and length of the floor, then subtract it by the length and width of the rug in order to find the area of the floor not covered by the rug.

 

(x+6)(x) = x²+6x (area of the floor)         (1)

(x+2)(x-6) = x²-8x-12 (area of the rug)   (2)

 

Now, simply (1) - (2), so x²+6x-x²+8x+12 = 14x+12