2. The outside of a closed glass display case measures 22 inches by 15 inches by 12 inches. The glass is $$ \frac{1}{2} $$ inch thick. How much air is contained in the case?

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To find the amount of air contained in the closed glass display case, we need to calculate the volume of the case and then subtract the volume of the glass itself.

### Step 1: Calculate the outer volume of the display case

The outer dimensions of the display case are given as:
- Length ($$L$$) = 22 inches
- Width ($$W$$) = 15 inches
- Height ($$H$$) = 12 inches

The formula for the volume of a rectangular prism is:
$$
V = L \times W \times H
$$

Substituting the values:
$$
V_{\text{outer}} = 22 \times 15 \times 12
$$

### Step 2: Calculate the inner dimensions of the display case

Since the glass is $$ \frac{1}{2} $$ inch thick, we need to subtract twice the thickness from each dimension (once for each side).

- Inner Length ($$L_i$$) = $$L - 2 \times \text{thickness} = 22 - 2 \times \frac{1}{2} = 22 - 1 = 21$$ inches
- Inner Width ($$W_i$$) = $$W - 2 \times \text{thickness} = 15 - 2 \times \frac{1}{2} = 15 - 1 = 14$$ inches
- Inner Height ($$H_i$$) = $$H - 2 \times \text{thickness} = 12 - 2 \times \frac{1}{2} = 12 - 1 = 11$$ inches

### Step 3: Calculate the inner volume of the display case

Now we can calculate the inner volume using the inner dimensions:
$$
V_{\text{inner}} = L_i \times W_i \times H_i = 21 \times 14 \times 11
$$

### Step 4: Calculate the volume of air contained in the case

The volume of air contained in the case is the difference between the outer volume and the inner volume:
$$
V_{\text{air}} = V_{\text{outer}} - V_{\text{inner}}
$$

Now, let's perform the calculations step by step.

1. Calculate $$V_{\text{outer}}$$
2. Calculate $$V_{\text{inner}}$$
3. Calculate $$V_{\text{air}}$$

Let's compute these values.
Calculate the value by following steps:
- step0: Calculate:
  $$22\times 15\times 12$$
- step1: Multiply the terms:
  $$330\times 12$$
- step2: Multiply the numbers:
  $$3960$$
Calculate or simplify the expression $$ 21 * 14 * 11 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$21\times 14\times 11$$
- step1: Multiply the terms:
  $$294\times 11$$
- step2: Multiply the numbers:
  $$3234$$
### Step 1: Calculate the outer volume
The outer volume of the display case is:
$$
V_{\text{outer}} = 22 \times 15 \times 12 = 3960 \text{ cubic inches}
$$

### Step 2: Calculate the inner volume
The inner volume of the display case is:
$$
V_{\text{inner}} = 21 \times 14 \times 11 = 3234 \text{ cubic inches}
$$

### Step 3: Calculate the volume of air contained in the case
Now, we can find the volume of air:
$$
V_{\text{air}} = V_{\text{outer}} - V_{\text{inner}} = 3960 - 3234 = 726 \text{ cubic inches}
$$

Thus, the amount of air contained in the display case is **726 cubic inches**.
The display case contains 726 cubic inches of air.

2. The outside of a closed glass display case measures 22 inches by 15 inches by 12 inches. The glass is \( \frac{1}{2} \) inch thick. How much air is contained in the case?

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