Problem $6(10pts).$ In the optimal solution to the chip manufacturing problem, Chip $1$ is not Your electronics company manufactures four types of computer chips, which sell for $$20,$ $$30,$

$$50,$ and $$40$ each. Each chip requires four separate steps to make: using raw materials to man$-$

ufacture the chips, etching microcircuits into the chips, laminating the chip, and testing it$.$ You

want to maximize profit for this month's production. You have $4000$ chips worth of raw material,

$600$ labor hours of etching time, $900$ labor hours of lamination time, and $700$ hours of testing time.

Taking into account all of the parameters, you set up the following problem, where $x_1,x_2,x_3,x_4$

represent how many of the four chips you produce.

maximize $20x_1+30x_2+50x_3+40x_4$

subject $to{x}_1+x_2+x_3+x_{4}4000$

$,0\mathrm{.}1x_1+0\mathrm{.}1x_2+0\mathrm{.}2x_3+0\mathrm{.}2x_{4}600$

$,0\mathrm{.}2x_1+0\mathrm{.}2x_2+0\mathrm{.}3x_3+0\mathrm{.}2x_{4}900$

$,0\mathrm{.}2x_1+0\mathrm{.}1x_2+0\mathrm{.}3x_3+0\mathrm{.}3x_{4}700$

$,x_1,x_2,x_3,x_{4}0$

efficient to produce. Suppose you redesign it so that its etching time changes by $t_1,$ its lamination

time changes by $t_2,$ and its testing time changes by $t_3.$ What values of $t_1,t_2,$ and $t_3$ preserve the

fact that Chip $1$ does not need to be produced to maximize profit?