If we let the shorter side be #bba# and the longer side be #bb(a+x)#, then perimeter is:
#2a+2(a+x)=46 \ \ \ \[1]#
and area is:
#a(a+x)=90 \ \ \ \[2]#
From #[1]#
#2a+2(x+a)=46#
#4a+2x=46#
#2x=46-4a=>x=23-2a#
Substituting in #[2]#
#a(a+23-2a)=90#
#a^2+23a-2a^2=90#
#-a^2+23a-90=0#
Factor:
#(5-a)(a-18)=0=>a=18 and a=5#
Substituting in #[1]#
#color(red)(a=5#
#2(5)+2(5+x)=46#
#10+10+2x=46=>color(red)(x=13#
#color(red)(a=18#
#2(18)+2(18+x)=46#
#72+2x=46=>color(red)(x=-13#
We now check these:
For #a=5, x=13#
#"area"=5(18)=90color(white)(888) sqrt#
#"perimeter"=10+36=46 color(white)(888)sqrt#
For #a=18, x=-13#
#"area"=18(5)=90color(white)(888)sqrt#
#"perimeter"=36+10=46color(white)(888)sqrt#
So the dimensions of the rectangle are:
#18xx5 "cm"#
