You are almost there, you just need more let:
let bmi=$total_weight/$total_height
Alternatives
There are so many ways to have an arithmetic context in a shell. The preferred standard way is the syntax $(( )):
total_weight=$(( $weight * 703 ))
expr (. ) , POSIX sh. ( $[ ], , .)
, . integer RHS :
declare -i weight height bmi total_weight total_height
total_weight=weight*703
total_height=height*height
bmi=total_weight/total_height
let.
(( )).
(( total_weight=weight*703 ))
(( total_height=height*height ))
(( bmi=total_weight/total_height ))
, expr - .
total_weight=$(expr $weight '*' 703)
total_height=$(expr $height '*' $height)
... meh, !
, bash . ( .)
, . , bc, awk .