Question

A toy manufacturer makes a toy truck, a toy car, and a toy boat. The production spreadsheet is:

Toy Time to Produce Cost to Produce Profit for Each Truck 10 minutes $1.00 $1.00 Car 12 minutes $0.75 $1.50 Boat 8 minutes $0.80 $0.60
After accounting for breaks, a worker actually works 400 minutes each day. The manufacturer needs each worker to generate a potential profit of $35 each day. Write a system of inequalities that expresses these constraints.

Answer 1

`\left\{\begin{array}{l}10x+12y+8z\le 400\\x+1.5y+0.6z\ge 35\end{array}\right.`

Explanation:

Let x be the number of toy trucks, y be the number of toy cars and z be the number of toy boats a worker makes each day.

1. If a worker spends 10 minutes to make one toy truck, then he spends 10x minutes to make x toy trucks. If a worker spends 12 minutes to make one toy car, then he spends 12y minutes to make y toy cars. If a worker spends 8 minutes to make one toy boat, then he spends 8z minutes to make z toy boats. In total he can spend at most 400 minutes each day, then

`10x+12y+8z\le 400.`

2. If a worker generates profit of $1.00 per one toy truck, then he generates profit of $x per x toy trucks. If a worker generates profit of $1.50 per one toy car, then he generates profit of $1.50y per y toy cars. If a worker generates profit of $0.60 per one toy boat, then he generates profit of $0.60z per z toy boats. The manufacturer needs each worker to generate a potential profit of $35 each day, then

`x+1.5y+0.6z\ge 35.`

Thus, the system of two inequalities is

`\left\{\begin{array}{l}10x+12y+8z\le 400\\x+1.5y+0.6z\ge 35\end{array}\right.`

Answer 2

Let x be the number of toy truck, y is the number of toy car, and z is the number of toy boat.

The constraint for the time

`10x+12y+8z\le 400`

The constraint for the profit

`x+1.50y+0.6z\ge 35`