Math Equation

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. 

Answer 1

x=length of the floor

x+6=width of the floor

(x+6)-4=width of the rug

[(x+6)-4]-8=length of the rug

simplify these expressions

x=length of the floor

x+6=width of the floor

x+2=width of the rug

x-6=length of the rug

the area of the floor minus the area of the rug gives the area of the uncovered floor

R(x)=area of floor-area of rug

R(x)=x(x+6)-(x-6)(x+2)

if you want to go further...

R(x)=x²+6x-x²+6x-2x+12

R(x)=10x+12

Answer 2

Find the area of the floor and the area of the rug and subtract them.  These will not be "nice" answers.  They will instead be expressions.

 

Area of the floor: if the length is x and the width is 6 feet greater, then the width can be thought of as x + 6.

 

A = lw, so A = x(x+6) = x² +6x

 

 

Area of the rug: the width  of the rug is 4 feet less than the width of the floor.  Since the width of the floor is x + 6, then the width of the rug is x + 6 - 4 = x + 2

 

the length of the rug is 8 feet less than the width of the rug, which we just said was x + 2. so the length of the rug is

x + 2 - 8 = x - 6.

 

Once again the A = lw, so (x -6)(x +2) = x² + 2x - 6x - 12 = x² - 4x -12

 

 

Subtracting the expression for the area of the floor and the area of the rug:

 

R(x) = x²+6x - (x² -4x - 12) 

which simplifies to 

 

R(x) = 10x +12

 

 

Answer 3

Natalie,

 

Let:      x = length of the floor (in ft).

   (x + 6) = width of the floor (in ft).

  [(x + 6) - 4] = (x + 2) = width of the rug which is 4 ft less than the width of the floor (in ft).

  [(x + 2) - 8] = (x - 6)  = length of the rug which is 8 ft less than the width of the rug (in ft).

   A_F =  x(x+6) = area of the floor which is its length times its width (in sq. ft). 

   A_R = (x - 6)(x + 2) = area of the rug which is its length times its width (in sq. ft.).

 

Then, the floor area that remains uncovered by the rug is:

A_F - A_R

= x(x+6) - (x - 6)(x + 2) 

= x² + 6x - (x² -4x -12)

= 10x + 12