Thread: pb solving algebra II 10th grade

This is what I have: One of the dolls that Dolls R Us manufactures is Talking Tommy (TT). Another doll without the talking mechanism is called SIlent Sally (SS). In one hour, the company can produce 8 TT or 20 SS. Because of the demand, the Cie knows that it must produce at least twice as many TT as SS dolls. The Cie spends no more that 48 hours per week making these two dolls. The profit on each TT is $3 and on SS is $7.50. How many of each dolls should be produces to maximize profit each week? What is the profit?
I have no idea of how to solve it. It is urgent. Thanks

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 $