Question

With the start of school approaching, a store is planning on having a sale on school materials. They have 600 notebooks and 400 pens in stock, and they plan on packing it in two different forms. In the package A, there will be 2 notebooks and 2 pens, and in the package B, there will be 3 notebooks and 1 pen. The price of the package A will be $6.50 and the price of the package B will be $7.00. How many packages should they put together of each type to obtain the maximum benefit?

Answer 1

150 of package A

100 of package B

Explanation:

Let x represent the number of package A.

Let y represent the number of package B.

Given that the price of package A will be $6.50 and the price of package B will be $7.00, the revenue (R) produced by selling the packages is given by the equation:

`R(x,y)=6.5x+7y`

To maximum the revenue, we first need to determine any constraints from the given information.

Constraint 1

There are 600 notebooks in stock. Since package A uses 2 notebooks and package B uses 3 notebooks, we need to ensure that:

• 
  `2x + 3y \leq 600`

Constraint 2

There are 400 pens in stock. Since package A uses 2 pens and package B uses 1 pen, we need to ensure that:

• 
  `2x + y \leq 400`

Constraints 3 & 4

Since we cannot have a negative number of packages, it must be that:

• 
  `x \geq 0`

• 
  `y \geq 0`

Therefore, we have four constraints represented by the following system of inequalities:

`\begin{cases}2x + 3y \leq 600\\2x + y \leq 400\\x \geq 0\\y \geq 0\end{cases}`

To maximize the revenue produced by selling the packages subject to the given constraints, we need to analyse the feasible region defined by the system of inequalities, which is the area where all the shaded regions defined by the inequalities overlap.

Upon graphing the inequalities (see attached), the feasible region has four vertices (corner points):

• (0, 0)
• (0, 200)
• (150, 100)
• (200, 0)

Determine the value of R at each vertex by substituting the x and y values of the points into the equation for R:

`\begin{aligned}(0, 0) \implies & R=6.5(0)+7(0)=\boxed{0}\\\\(0, 200) \implies& R=6.5(0)+7(200)=\boxed{1400}\\\\(150, 100) \implies & R=6.5(150)+7(100)=\boxed{1675}\\\\(200, 0) \implies & R=6.5(200)+7(0)=\boxed{1300}\end{aligned}`

So, the maximum value of R is $1,675 at vertex (150, 100).

This means that to maximum their profit, the school should put together:

• 150 of package A
• 100 of package B