Alice has started working at a telecommunications company called Ignite. This company is planning to invest in machine$-$to$-$machine technology, which allows devices to talk and interact with each other directly without human input. They begin by producing modems and routers, which require designing, building and testing.

$-$Task A$-$

Your task is to recommend how many modems and routers they should produce each week to maximise the income with the constraints detailed below.

$$ The company can only produce a maximum of $50$ modems and $60$ routers each week.

$$ The company has a total of $130$ total design hours per week. Each modem takes $1$ hour to design, and each router takes $2$ hours to design.

$$ The company has a total of $80$ total build hours per week. Each modem takes $1$ hour to build, and each router takes $1$ hour to build.

$$ The company has a total of $80$ total test hours per week. Each modem takes $90$ minutes to test, and each router takes $30$ minutes to test.

$$ Each modem brings in a profit of $$400,$ and each router brings in a profit of $$600.$

Recommend to the company the number of modems and routers that they should produce each week to maximise the income.

$-$Task B$-$

One of Alice$$s colleagues suggests that they could make a profit of $$525$ on each modem and a profit of $$525$ on each router.

Using these new prices, recommend to the company the number of modems and routers that they should produce to maximise their income, and recommend whether they should switch to the colleague$$s new profit structure.

$-$Key Points$-$

Task A

$$ Write the constraints and state clearly what each constraint means.

$$ Draw the feasible region.

$$ Find all vertices on the feasible region.

$$ Use vertices to calculate and optimise the profit.

$$ Write a conclusion.

Task B

$$ Derive the new profit equation.

$$ Calculate the gradient of the new profit equation.

$$ Use vertices to calculate and optimise the profit with new profit equation.

$$ Calculate the gradient between optimal points.

$$ Write a conclusion commenting on the relationship between the gradient of the new profit equation and the gradient between the optimal points.