A utility shelf in a warehouse is used to store containers of paint and containers of varnish. Containers of paint weight 50 pounds each and containers of varnish weight 35 poinds each. The self can hold up to 32 containers, the combined weith of which must not exceed 1,450 pounds. Let x be the number of containers of paint and y be the number of container of varinish. Which of the following systems of inequalities represents this relationship ?

Options

A. `50x + 35yle 32`
`x +y le 1,450`
B. `50x + 35y le 1,450`
`x +y le 32`
C. `85 (x+y) le 1,450`
`x +y le 32`
D. `50x + 35y y le 1,450`
`x +y le 85`

Explanation

To solve the problem, we need to establish a system of inequalities based on the constraints provided for the utility shelf in the warehouse.

### Step 1: Define the Variables
Let:
- \( x \) = number of containers of paint
- \( y \) = number of containers of varnish

### Step 2: Establish the Container Limit Inequality
The shelf can hold a maximum of 32 containers in total. Therefore, we can express this constraint as:
\[ x + y \leq 32 \]

### Step 3: Establish the Weight Limit Inequality
Each container of paint weighs 50 pounds, and each container of varnish weighs 35 pounds. The combined weight of the containers must not exceed 1,450 pounds. This can be expressed as:
\[ 50x + 35y \leq 1450 \]

### Step 4: Write the System of Inequalities
Combining the two inequalities we derived, we get the following system of inequalities:
1. \( x + y \leq 32 \)
2. \( 50x + 35y \leq 1450 \)

### Conclusion
Thus, the system of inequalities that represents the relationship is:
\[ \begin{cases} x + y \leq 32 \\ 50x + 35y \leq 1450 \end{cases} \]