first let us translate the given detail to some clear mathematical expressions:

# of produced TT dolls = TT

# of produced SS dolls = SS

"must produce at least twice as many TT as SS dolls" ----> TT >= 2*SS

"The Cie spends no more that 48 hours per week making these two dolls" and

"In one hour, the company can produce 8 TT or 20 SS"--->

TT/8 + SS/20 <= 48

"The profit on each TT is $3 and on SS is $7.50" ---> P = 3*TT + 7.5*SS

so in front of us is a optimization problem subjected to 2 restrictions:

P = 3*TT + 7.5*SS

TT >= 2*SS

TT/8 + SS/20 <= 48

grad(P) = [3, 7.5] != 0 therefore the only extrema points of P can be found on corner points of the perimeter defined by:

TT >= 2*SS

TT/8 + SS/20 <= 48

one can easily verify that it's a triangle with corners at(coordinates [SS, TT]):

(0,0), (0,384), (160,320)

(0,0) minimizes the profit & (160,320) maximizes it so the cie must produce

TT = 320 and SS = 160

and the resulting profit will be 2160 $