Questions: A candy shop puts together two prepackaged assortments to be given to trick-or-treaters on Halloween. Assortment A contains 2 candy bars and 2 suckers and yields a profit of 40 cents. Assortment B contains 1 candy bar and 2 suckers and yields a profit of 30 cents. The store has available 200 candy bars and 300 suckers. Use this information to complete parts (a) through (e) below. (c) Give the inequalities that x and y must satisfy because x and y cannot be negative. y ≥ x ≥

A candy shop puts together two prepackaged assortments to be given to trick-or-treaters on Halloween. Assortment A contains 2 candy bars and 2 suckers and yields a profit of 40 cents. Assortment B contains 1 candy bar and 2 suckers and yields a profit of 30 cents. The store has available 200 candy bars and 300 suckers. Use this information to complete parts (a) through (e) below.

(c) Give the inequalities that x and y must satisfy because x and y cannot be negative.

y ≥
x ≥

Transcript text: A candy shop puts together two prepackaged assortments to be given to trick-or-treaters on Halloween. Assortment A contains 2 candy bars and 2 suckers and yields a profit of 40 cents. Assortment B contains 1 candy bar and 2 suckers and yields a profit of 30 cents. The store has available 200 candy bars and 300 suckers. Use this information to complete parts (a) through (e) below. (c) Give the inequalities that x and y must satisfy because x and y cannot be negative. \[ \left\{\begin{array}{l} y \geq \square \\ x \geq \square \end{array}\right. \]

Solution

Solution Steps

Step 1: Formulate the Objective Function

To maximize the total profit, we need to solve the following objective function: $$Z = 0.4x + 0.3y$$ where:

$x$ and $y$ are the quantities of assortments A and B to produce, respectively.
$p_A = 0.4$ and $p_B = 0.3$ are the profits from selling one package of assortment A and B, respectively.

Step 2: Set Up the Constraints

The constraints for this problem are:

Candy bars: $a_A \cdot x + a_B \cdot y \leq C_{candy}$, where $a_A = 2$, $a_B = 1$, and $C_{candy} = 200$.
Suckers: $b_A \cdot x + b_B \cdot y \leq C_{suckers}$, where $b_A = 2$, $b_B = 2$, and $C_{suckers} = 300$.
Non-negativity: $x \geq 0$ and $y \geq 0$.

Step 3: Solve the Linear Programming Problem

Using linear programming, we find the optimal solution to be $x = 50$ and $y = 100$, with a total profit of $Z = 50$.