Linear Programming Worksheet + Answers - Free Printable

Let's solve this linear programming problem step by step.

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Problem Summary:

A potter makes cups and plates.
We are to maximize her profit, given constraints on time and clay.

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Step 1: Define Variables

- Let $ x $ = number of cups
- Let $ y $ = number of plates

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Step 2: Objective Function (Profit)

She makes:
- $ \$2 $ profit per cup → $ 2x $
- $ \$1.50 $ profit per plate → $ 1.5y $

So, total profit:
$$
P = 2x + 1.5y
$$
We want to maximize $ P $.

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Step 3: Constraints

#### Time Constraint
- Time to make a cup: 6 minutes
- Time to make a plate: 3 minutes
- Total time available: 20 hours = $ 20 \times 60 = 1200 $ minutes

So:
$$
6x + 3y \leq 1200
$$

Divide entire inequality by 3:
$$
2x + y \leq 400 \quad \text{(simplified)}
$$

But in the image, it says:
> $ 6x + 3y \leq 1200 $ — this is correct.

#### Clay Constraint
- Each cup uses $ \frac{3}{4} $ lb of clay
- Each plate uses 1 lb of clay
- Total clay available: 250 lbs

So:
$$
\frac{3}{4}x + 1y \leq 250
$$

Multiply through by 4 to eliminate fraction:
$$
3x + 4y \leq 1000
$$

But in the image, it says:
> $ 0.75x + y \leq 250 $ — which is equivalent to $ \frac{3}{4}x + y \leq 250 $ — so that’s correct.

#### Non-negativity
$$
x \geq 0, \quad y \geq 0
$$

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Constraints Recap:

1. $ 6x + 3y \leq 1200 $ → Time
2. $ 0.75x + y \leq 250 $ → Clay
3. $ x \geq 0 $
4. $ y \geq 0 $

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Step 4: Graph the Feasible Region

We are told the graph has corners at:
- $ (0, 250) $
- $ (120, 160) $
- $ (200, 0) $

Wait — let's verify these points and find all corner points.

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Find Intersection Points

We'll find the corner points by solving the system of equations.

#### Constraint 1: $ 6x + 3y = 1200 $
Simplify: divide by 3 → $ 2x + y = 400 $ → $ y = 400 - 2x $

#### Constraint 2: $ 0.75x + y = 250 $ → $ y = 250 - 0.75x $

Now, find intersection of the two lines:

Set equal:
$$
400 - 2x = 250 - 0.75x
$$

Solve:
$$
400 - 250 = 2x - 0.75x \\
150 = 1.25x \\
x = \frac{150}{1.25} = 120
$$

Then:
$$
y = 400 - 2(120) = 400 - 240 = 160
$$

So, intersection point: $ (120, 160) $

Now find intercepts:

#### From constraint 1: $ 6x + 3y = 1200 $

- When $ x = 0 $: $ 3y = 1200 $ → $ y = 400 $
- But check if this satisfies clay constraint:
$ 0.75(0) + y = y \leq 250 $ → $ y \leq 250 $, so $ y = 400 $ is not feasible

So, the actual feasible region is bounded by both constraints.

#### Intercepts for clay constraint: $ 0.75x + y = 250 $

- $ x = 0 $: $ y = 250 $ → point $ (0, 250) $
- $ y = 0 $: $ 0.75x = 250 $ → $ x = \frac{250}{0.75} = \frac{250 \times 4}{3} = \frac{1000}{3} \approx 333.33 $

But check time constraint at $ (333.33, 0) $:
$ 6(333.33) + 3(0) = 2000 $ minutes > 1200 → not feasible

So we need to find where the clay line hits the time constraint.

We already found the intersection: $ (120, 160) $

Now find where time constraint hits axes within feasible region:

- $ 6x + 3y = 1200 $
- $ x = 0 $: $ 3y = 1200 $ → $ y = 400 $ → but clay limit is only 250 → so not feasible
- $ y = 0 $: $ 6x = 1200 $ → $ x = 200 $

Check clay at $ (200, 0) $:
$ 0.75(200) + 0 = 150 \leq 250 $ → feasible

So $ (200, 0) $ is a corner point.

Now, $ (0, 250) $: check time
$ 6(0) + 3(250) = 750 \leq 1200 $ → feasible

So the three corner points of the feasible region are:

1. $ (0, 250) $ — from clay constraint at $ x=0 $
2. $ (120, 160) $ — intersection of both constraints
3. $ (200, 0) $ — from time constraint at $ y=0 $

These match the graph provided.

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Step 5: Evaluate Profit at Each Corner Point

Recall: $ P = 2x + 1.5y $

1. At $ (0, 250) $:
$$
P = 2(0) + 1.5(250) = 0 + 375 = 375
$$

2. At $ (120, 160) $:
$$
P = 2(120) + 1.5(160) = 240 + 240 = 480
$$

3. At $ (200, 0) $:
$$
P = 2(200) + 1.5(0) = 400 + 0 = 400
$$

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Step 6: Find Maximum Profit

Compare:
- $ (0, 250) $: $ P = 375 $
- $ (120, 160) $: $ P = 480 $
- $ (200, 0) $: $ P = 400 $

✔ Maximum profit is $ 480 $ at $ (120, 160) $

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✔ Final Answer:

To maximize profit, the potter should make:

- 120 cups
- 160 plates

This gives a maximum profit of \$480.

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✔ Verification of Constraints:

1. Time:
$ 6(120) + 3(160) = 720 + 480 = 1200 $ minutes → ✔️ (exactly 20 hours)

2. Clay:
$ 0.75(120) + 1(160) = 90 + 160 = 250 $ lbs → ✔️ (exactly 250 lbs)

Both constraints are satisfied with equality — optimal point lies at the intersection.

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✔ Conclusion:

Make 120 cups and 160 plates to maximize profit at \$480.