Abey Numkena is an interior designer. She has been asked to locate an oriental rug for a new corporate office. As a rule, the rug should cover 1/2 of the total floor area with a uniform width surrounding the rug.
a. If the dimensions of the room are 12 feet by 16 feet, write an equation to model the situation.
b. Graph the related function.
c. What are the dimensions of the rug?

Question

Abey Numkena is an interior designer. She has been asked to locate an oriental rug for a new corporate office. As a rule, the rug should cover 1/2 of the total floor area with a uniform width surrounding the rug.
a. If the dimensions of the room are 12 feet by 16 feet, write an equation to model the situation.
b. Graph the related function.
c. What are the dimensions of the rug?

anonymous · Answer

We know the room is 12x16 so $$A_R= 12 \times 16 = 192ft^{2}$$ the rug must therefore have an area of $$A_r=\frac{A_R}{2} = \frac{192}{2} = 96ft^{2}$$ We also know that there must be a uniform gap, call it 'x' all the way around: |dw:1417200466946:dw| so: $$A_r=16-2x \times 12-2x $$ or $$96= (16-2x) \times (12-3x)$$  now solve ofor x and back-compute the rug size.