A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?

Options

A. `0 lt x le 10`
B. `0 lt x le 11 2/3`
C. `0 lt x le 17 1/2`
D. `0 lt x le 20`

Explanation

To solve the problem step by step, we need to derive the inequality that represents the allowable width \( x \) of the box based on the given conditions.

### Step 1: Understand the Problem
We know that the sum of the perimeter of the base of the box and the height must not exceed 130 inches. The box has a height of 60 inches, and the length \( L \) is 2.5 times the width \( W \).

### Step 2: Write the Perimeter Formula
The perimeter \( P \) of the base of a rectangular prism (box) is given by:
\[ P = 2L + 2W \]
However, since we are interested in the sum of the perimeter and the height, we can express this as:
\[ P + H \leq 130 \]
where \( H \) is the height of the box.

### Step 3: Substitute the Height
Given that the height \( H = 60 \) inches, we can substitute this value into the inequality:
\[ 2L + 2W + 60 \leq 130 \]

### Step 4: Simplify the Inequality
Now, we simplify the inequality:
\[ 2L + 2W \leq 130 - 60 \]
\[ 2L + 2W \leq 70 \]

### Step 5: Divide by 2
To make the equation simpler, we can divide the entire inequality by 2:
\[ L + W \leq 35 \]

### Step 6: Substitute Length in Terms of Width
Since we know that the length \( L \) is 2.5 times the width \( W \), we can substitute \( L \) in the inequality:
\[ 2.5W + W \leq 35 \]

### Step 7: Combine Like Terms
Combining the terms gives us:
\[ 3.5W \leq 35 \]

### Step 8: Solve for Width
Now, we solve for \( W \):
\[ W \leq \frac{35}{3.5} \]
Calculating this gives:
\[ W \leq 10 \]

### Step 9: Define the Allowable Width
Since width cannot be negative, we also have the condition:
\[ W > 0 \]
Thus, we can express the allowable width \( x \) (where \( x \) represents the width) as:
\[ 0 < x \leq 10 \]

### Final Inequality
The final inequality that shows the allowable width \( x \) of the box is:
\[ 0 < x \leq 10 \]