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There are several elevators in the Sandy Beach Hotel. Each elevator can only hold 1600 pounds max due to safety reasons, as well as hold only 12 people. Let's assume the average weight of an adult and child are 175 and 70 pounds, respectively. Let's also include 150 pounds of baggage, and an additional 10 pounds of carry on for each person.

Determine the variables.

5 answers

To define the variables related to the constraints of the Sandy Beach Hotel's elevators, we will consider the weight limits and passenger limits. Here are the variables we can establish:

  1. Weight Limit Variable:

    • Maximum weight that the elevator can hold: \( W_{max} = 1600 \) pounds.

  2. People Limit Variable:

    • Maximum number of individuals that can ride in the elevator: \( N_{max} = 12 \) people.

  3. Weight Variables:

    • Average weight of an adult: \( W_a = 175 \) pounds.
    • Average weight of a child: \( W_c = 70 \) pounds.
    • Average baggage weight per person: \( W_b = 150 \) pounds.
    • Average carry-on weight per person: \( W_{co} = 10 \) pounds.

  4. Passenger Count Variables:

    • Let \( A \) = number of adults.
    • Let \( C \) = number of children.

  5. Total Weight Calculation:

    • The total weight of adults: \( W_A = A \times W_a \).
    • The total weight of children: \( W_C = C \times W_c \).
    • Total baggage weight for all passengers: \( W_B = (A + C) \times W_b \).
    • Total carry-on weight for all passengers: \( W_{CO} = (A + C) \times W_{co} \).

  6. Total Weight Formula:

    • Total weight in the elevator:
      \[ W_{total} = W_A + W_C + W_B + W_{CO} = A \times W_a + C \times W_c + (A + C) \times W_b + (A + C) \times W_{co}. \]

In summary, the key variables are:

  • \( A \) (number of adults)
  • \( C \) (number of children)
  • \( W_{max} \) (maximum weight limit of the elevator)
  • \( N_{max} \) (maximum number of people in the elevator)
  • \( W_a \), \( W_c \), \( W_b \), \( W_{co} \) (weights associated with adults, children, baggage, and carry-ons respectively).
Sadly we can only use two...

If we can only use two variables to represent the constraints of the Sandy Beach Hotel elevators, we can choose the following:

  1. Variable 1: Total Number of Passengers (P)
    This variable (P) will represent the total number of individuals in the elevator, including both adults and children. \[ P = A + C \] where \( A \) is the number of adults and \( C \) is the number of children.

  2. Variable 2: Total Weight (W)
    This variable (W) will represent the total weight of all passengers, their baggage, and carry-ons. \[ W = (A \times 175) + (C \times 70) + (P \times 150) + (P \times 10) \] Simplifying this, we can express it as: \[ W = (A \times 175) + (C \times 70) + (P \times 160) \] where the component \( P \times 160 \) accounts for both baggage and carry-on per person (150 + 10).

Summary of Variables:

  1. \( P \): Total number of passengers (adults + children).
  2. \( W \): Total weight in pounds.

These two variables can be used to enforce the elevator's maximum capacity constraints:

  • The maximum number of passengers: \( P \leq 12 \)
  • The maximum weight limit: \( W \leq 1600 \) pounds.
Go ahead and make a system of linear equations.

To create a system of linear equations using the two variables \( P \) (total number of passengers) and \( W \) (total weight), we can derive equations based on the constraints provided.

Variables:

  1. \( P = A + C \) (Total number of passengers)
  2. \( W = (A \times 175) + (C \times 70) + (P \times 160) \) (Total weight)

Constraints:

  1. Passenger Limit: The total number of passengers must not exceed 12. \[ P \leq 12 \]

  2. Weight Limit: The total weight must not exceed 1600 pounds. \[ W \leq 1600 \]

System of Linear Equations:

To express this as a system of equations and inequalities, we can represent the constraints as follows:

  1. From the passenger limit:
    \[ P - 12 = 0 \] (This can be treated as an equation, with equality when at maximum capacity.)

  2. From the weight limit:
    \[ W - 1600 = 0 \] (This can also be treated as an equation, with equality when at maximum weight capacity.)

Final System of Equations:

Thus, the system of linear equations can be written as: \[ \begin{align*}

  1. & \quad P = A + C \
  2. & \quad W = 175A + 70C + 160P \
  3. & \quad P - 12 = 0 \
  4. & \quad W - 1600 = 0 \ \end{align*} \]

Interpretation:

  • Equation 1 represents the relationship between the number of adults, children, and total passengers.
  • Equation 2 models the total weight based on the number of adults and children.
  • Equation 3 ensures that the total number of passengers does not exceed 12.
  • Equation 4 ensures that the total weight does not exceed 1600 pounds.

You can solve this system by substituting \( P \) and \( W \) with their respective equations based on values for \( A \) (number of adults) and \( C \) (number of children). This will allow you to analyze permissible allocations of adults and children based on the constraints.