Question

Josh has 8 days to make pots and plates to sell at a local fair. Each pot weighs 2 pounds and each plate weighs 1 pound. Josh cannot carry more than 50 pounds to the fair. Each day, he can make at most 5 plates and at most 3 pots. He will make $12 profit for every plate and $25 profit for every pot that he sells.
a. Write linear inequalities to represent the number of pots p and plates a Josh may bring to the fair.
b. List the coordinates of the vertices of the feasible region.
c. How many pots and how many plates should Josh make to maximize his potential profit?

Answer 1

The optimal number of pots and plates Josh should make is given from

the graph of the inequalities.

Response:

a. 2·p + a ≤ 50

p ≤ 24

a ≤ 40

b. The coordinates of the vertices of the feasible region are; (24, 2), (40, 5), (0, 40), (24, 0), (0, 0)

c. 40 pots and 5 plates

How can the optimal number of plates and pots Josh should make be found?

Given:

Number of days Josh has to make pots and plates to sell = 8 days

Weight of each pot = 2 pounds

Weight of each plate = 1 pound

Weight Josh cannot carry = More than 50 pounds

Number plates he can make each da = At most 5 plates

Number of pots he can make each day = At most 5 pots

The profit Josh makes for each plate sold = $12

Profit from each pot sold = $25

a. Let p represent the number of pots Josh makes and let a represent the

number plates. The linear inequalities are;

2·p + a ≤ 50

p ≤ 8 × 3 = 24

p ≤ 24

a ≤ 5 × 8 = 40

a ≤ 40

b. When p = 24, we have;

2 × 24 + a = 50

a = 2

When a = 40, 2·p + 40 = 50, gives;

p = 5

The vertex points are;

The coordinates of the vertices of the feasible region are;

(24, 2), (40, 5), (0, 40), (24, 0), (0, 0)

c. The profit function is; P = 25·p + 12·a

The profit at the vertices are;

$\begin{tabular}{|c|c|c|}p&a&Profit\\0&0&0\\0&40&12 \times 40 = 480\\24&0&25 \times 24 = 600\\24&2&12 \times 2 + 25 \times 40 = 624\\40&5&12 \times 5 + 25 \times 40 = 660\end{array}\right]$

To maximize his potential profit, therefore;

• Josh should make 40 pots and 5 plates.

Learn more about linear optimization here:

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