Question

Two shops makes the same kind of garden shed. The preparation time and assembly time (hours) for Shop A is 5 hour and 15 hours respectively to make one shed while Shop B times are 10 hours and 7.5 hours. For the two shops combined the manufacturer can afford to use up to 200 hours for preparation and 240 hour for assembly for per week. Shop A earns a profit of $350 per shed and Shop B earns a profit of $380 per shed. How many sheds per week should the manufacturer make in each shop to maximize profit?

Answer 1

Create a system of linear inequalities.

`\begin{gathered} 5x+10y\le200 \\ 15x+7.5y\le240 \end{gathered}`

Multiply both sides of the first inequality by 3. Thus, we obtain the following:

`\begin{gathered} 15x+30y\le600 \\ 15x+7.5y\le240 \end{gathered}`

Subtract the second inequality from the first inequality.

`\begin{gathered} 22.5y\le360 \\ y\le16 \end{gathered}`

To obtain the value of x, substitute 16 in the first inequality.

`\begin{gathered} 5x+10y\le200 \\ 5x+10(16)\le200 \\ 5x+160\le200 \\ 5x\le40 \\ x\le8 \end{gathered}`

Solve for the number of hours per shop.

`\begin{gathered} A\colon5x+15x=20x=20(8)=160\text{ hours} \\ B\colon10y+7.5y=17.5y=17.5(16)=280\text{hours} \end{gathered}`

Thus,

`\begin{gathered} ShopA\colon8 \\ ShopB\colon16 \end{gathered}`