Question

~~~30 POINTS~~~ Please answer this math question!

Answer 1

With a dilation, each dimension increases by the factor.  Thus, if we let the dimensions be x and y, the new dimensions are 2x and 2y.

(a): The original perimeter is 2(x+y), but the new one is 2(2x+2y).  This is twice the original perimeter, so it is 18*2=36.

(b): The original area is xy, and the new one is (2x)(2y), or 4xy.  This is four times the original area, or 20*4=80.

(c): As it's given that the side lengths are integers, the intended solution is most likely to divide by 2 in the perimeter to see that the sum of the side-lengths is 9 and their product is 20.  Guessing/checking values for each side, we see that 4 and 5 work for the smaller rectangle.  Multiplying by two, the larger one has lengths 8 and 10.

Alternatively, we set them to x and y and use the equations:
x+y=9
xy=20

Dividing by y, we see that x=20/y.  Substituting, we have that y+20/y=9.  Subtracting 9 and multiplying by y, we have:

y^2-9y=20

Factoring, we have (y-5)(y-4)=0.  The solutions to this equation are 4 and 5, which result in x=5, y=4 or x=4, y=5 respectively.  Thus, we see that 4 and 5 are the side-lengths.  Note that this solution did not require the assumption that the side-lengths are integers!