The Partside Packing Company needs a rectangular shipping box. The box must have a length of 1 foot, a width of 8 inches, and a volume of at least 700 cubic inches. What is the least number of inches in height of the box such that the height is a whole number?

Answer

To solve the problem, we need to find the least height of a rectangular shipping box that meets the specified volume requirement. Let's break down the solution step by step.

### Step 1: Convert Length to Inches
The length of the box is given as 1 foot. We need to convert this measurement into inches since the width and volume are given in inches.

- **Conversion**:
\[ \text{Length} = 1 \text{ foot} = 12 \text{ inches} \]

### Step 2: Write Down the Known Values
Now that we have the length in inches, we can summarize the known values:
- Length (L) = 12 inches
- Width (W) = 8 inches
- Volume (V) ≥ 700 cubic inches

### Step 3: Use the Volume Formula
The volume \( V \) of a rectangular box (cuboid) is calculated using the formula:
\[ V = L \times W \times H \]
where \( H \) is the height of the box.

### Step 4: Set Up the Inequality
Since we want the volume to be at least 700 cubic inches, we can set up the following inequality:
\[ 12 \times 8 \times H \geq 700 \]

### Step 5: Calculate the Product of Length and Width
Now, calculate the product of the length and width:
\[ 12 \times 8 = 96 \]
So, we can rewrite the inequality as:
\[ 96H \geq 700 \]

### Step 6: Solve for Height (H)
To find \( H \), divide both sides of the inequality by 96:
\[ H \geq \frac{700}{96} \]
Calculating the right side:
\[ H \geq 7.29166667 \]

### Step 7: Find the Least Whole Number
Since we need the height \( H \) to be a whole number, we take the smallest whole number that is greater than or equal to 7.29166667, which is 8.

### Final Answer
Thus, the least height of the box that meets the requirements is:
\[ \text{Height} = 8 \text{ inches} \]

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