Question

a floor manager is going to install two types of machine, small and large. the following table shows the number of operators and the space requirements for each machine:

Small Large Maximum available
Number of operators 5 4 40
Space in m2 2 4 28

i) Taking x to represent the number of small machines and y to represent the number of large machines, write down two inequalities in x and y and illustrate these on a graph.

ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, calculate the number of each type of machine that should be installed in order to have a maximum profit. What is the profit?

Answer 1

Answer on Question #49450 – Math – Linear Algebra

A floor manager is going to install two types of machine, small and large. The following table shows the number of operators and the space requirements for each machine:

i) Taking $x$ to represent the number of small machines and $y$ to represent the number of large machines, write down two inequalities in $x$ and $y$ and illustrate these on a graph.

ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, calculate the number of each type of machine that should be installed in order to have a maximum profit. What is the profit?

Solution

i) Let $x$ represent the number of small machines and $y$ represent the number of large machines, the two inequalities in $x$ and $y$ will be the following:

$$\begin{array}{l} 5x + 4y \leq 40 \\ 2x + 4y \leq 28 \\ \end{array}$$

It is obvious that from practical point of view, we require that $x \geq 0$, $y \geq 0$.

Straight line $5x + 4y = 40$ is blue in the figure below, its x-intercept is $(x, y) = (8,0)$, its y-intercept is $(x, y) = (0,10)$.

Straight line $2x + 4y = 28$ is yellow in the figure below, its x-intercept is $(x, y) = (14,0)$, its y-intercept is $(x, y) = (0,7)$.

Two straight lines intersect at the point where $x = 4, y = 5$.

ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, to calculate the number of each type of machine that should be installed in order to have a maximum profit

Method 1

Check values of function $120x + 200y$ under different cutting-edge points of region

$$\{5x + 4y \leq 40, 2x + 4y \leq 28, x \geq 0, y \geq 0\}$$ $$(x, y) = (8, 0), (x, y) = (0, 7), (x, y) = (4, 5)$$

If $(x,y) = (8,0)$, then $120x + 200y = 120*8 = 960$ Euros

If $(x,y) = (0,7)$, then $120x + 200y = 200*7 = 1400$ Euros

If $(x,y) = (4,5)$, then $120x + 200y = 120*4 + 200*5 = 1480$ Euros

The maximum value of function $120x + 200y$ in the region $\{5x + 4y \leq 40, 2x + 4y \leq 28, x \geq 0, y \geq 0\}$ is attained when $(x,y) = (4,5)$, it equals 1480 Euros.

Method 2

We use solver in excel.

The target cell is 120x + 200y = max

Restrictions are:

- 5x + 4y <= 40

- 2x + 4y <= 28

(x and y are empty cells)

The maximum profit is: 1480 Euros (when x=4, y=5).

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