we can solve this by setting up 2 equations and finding a quadratic equation that relates to both equations. the perimeter of a rectangle is 2*base +2*height = perimeter. and area = b*h so 2b+2h=16(perimeter) and b*h=15(area). solving the area equation for b gives us b=15/h. substitute 15/h for b in the perimeter equation gives 2(15/h)+2h=16 so 30/h+2h=16 multiply the equation by h gives
30+2h^2=16h. manipulate the equation so the resulting quadratic is obvious (just set one side 0) and rearrange terms. so then we have 2h^2-16h+30=0. factor out the 2 to make the quadratic easier to solve ie..2(h^2-8h+15)=0. now solving the quadratic gives 2(h-5)(h-3)=0 so h=3 or h=5 just pick one and substitute it for h in the perimeter equation. choosing 5 for h we get 2b+2(5)=16 which gives 2b+10=16 which gives 2b=6 so b = 3. with b=3 and h=5 for each equation we get 2(b)+2(h)=16 = 2(3)+2(5) =16 this is correct. substituting the values into the area equation gives b*h=15 = 3*5=15 this also checks out so the dimensions of the rug are 3x5.