Solve the following word problem. You should get infinite solutions. So find the pattern for them and give a range of $t$ values that is possible based on the word problem.

A trucking company plans on buying $4$ kinds of trucks: a pick$-$up$,$ a small flatbed, a large flatbed, and a dump truck. The pick$-$up needs one worker to operate, $4$ hours of maintenance and $3$ hours of cleaning. The small flatbed needs one worker to operate, $1$ hour of maintenance and $1$ hour of cleaning. The large flatbed needs one worker to operate, $3$ hours of maintenance and $2$ hours of cleaning. The dump truck needs $2$ workers to operate, $3$ hours of maintenance and $2$ hours of cleaning. If the compariy has exactly $46$ workers, exactly $140$ hours for maintenance and exactly $112$ hours for cleaning. How many of each kind of truck should they purchase?

$102$

The Kimble Firefighting Equipment Company makes alarm valves. It has two assembly plants: Plant A in Atlanta and Plant B in Buffalo. All the alarm valves produced are shipped to two warehouse locations, one in Pittsburgh and one in Savannah. The grid below shows the distance between these locations:

$\backslash$table$[[,$Warehouse I: Pittsburgh,Warehouse I: Savannah$],[$Plant A in Atlanta,$685$ miles,$248$ miles$],[$Plant B in Buffalo,$215$ miles,$907$ miles$]]$

a$.$ The current cost per mile for each alarm to ship is $$0\mathrm{.}03.$ Calculate the cost to ship an alarm valve from each location to each warehouse. Put your answers into a table like the one above.

Plant A can produce at most $8000$ alarm valves and Plant B can produce at most $10000$ alarm valves each week. The cost to produce an alarm valve is $$12$ at plant A and $$18$ at Plant $B.$ The current weekly demand is for at least $9000$ alarm valves to be shipped to Warehouse I and at least $6000$ alarm valves to be shipped to Warehouse II$.$

b$.$ Write a linear programming problem to minimize the total weekly cost of production and shipment for the alarm valves made in each plant and housed in each warehouse.

$($Hint: There should be $4$ variables for example let $x$ be the alarm valves produced in Atlanta and housed in Pittsburgh in a week, $y$ be the alarm valves produced in Atlanta and housed in Savannah. $q,$ and so on$.)$

Define your variables, state the objective function and list all constraints.

c$.$ Solve the problem using the simplex solver. State your solution in a complete sentence.

d$.$ In winter diesel gas prices increase the cost to ship each alarm has increased from $$0\mathrm{.}03$ to $$0\mathrm{.}05$ per mile. What is the impact on your weekly costs?

e$.$ Rewrite your LP $($linear programming problem$)$ with these new costs and solve.

f$.$ The plant in Buffalo is old and needs major renovations, while the plant in Atlanta is newer and has room to expand. The company wants to look at the impact of shutting down the Buffalo plant using the same costs from problem d$.($with the higher gas prices$).$

First expand the capacity of the Atlanta plant from $8000$ to $16000$ and take the capacity of the Buffalo plant to $0.$ Write the new LP and the optimal solution.

What would be the cost of shutting down the Buffalo plant? Would you recommend shutting down the Buffalo plant? Use your analysis to justify your answer