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Sagot :
Certainly! Let's go through the steps to determine how many cubic inches of flour will be saved by not filling the box to the top.
### Step 1: Determine Total Volume of the Box
First, let's calculate the total volume if the box were filled completely. Given the dimensions:
- Height = 12 inches
- Width = 8 inches
- Depth = 3 inches
The volume, [tex]\( V \)[/tex], is calculated by multiplying these dimensions together:
[tex]\[ V_{\text{total}} = \text{height} \times \text{width} \times \text{depth} \][/tex]
[tex]\[ V_{\text{total}} = 12 \, \text{inches} \times 8 \, \text{inches} \times 3 \, \text{inches} \][/tex]
[tex]\[ V_{\text{total}} = 288 \, \text{cubic inches} \][/tex]
### Step 2: Determine Fill Height
Next, the manufacturer wants to leave 1.5 inches unfilled at the top of the box. Thus, the filling height, [tex]\( H_{\text{fill}} \)[/tex], would be:
[tex]\[ H_{\text{fill}} = \text{height} - 1.5 \, \text{inches} \][/tex]
[tex]\[ H_{\text{fill}} = 12 \, \text{inches} - 1.5 \, \text{inches} \][/tex]
[tex]\[ H_{\text{fill}} = 10.5 \, \text{inches} \][/tex]
### Step 3: Calculate the Filled Volume
The volume when the box is filled up to 1.5 inches from the top is calculated similarly, using the fill height:
[tex]\[ V_{\text{fill}} = H_{\text{fill}} \times \text{width} \times \text{depth} \][/tex]
[tex]\[ V_{\text{fill}} = 10.5 \, \text{inches} \times 8 \, \text{inches} \times 3 \, \text{inches} \][/tex]
[tex]\[ V_{\text{fill}} = 10.5 \times 8 \times 3 \][/tex]
[tex]\[ V_{\text{fill}} = 252 \, \text{cubic inches} \][/tex]
### Step 4: Calculate the Saved Volume
Finally, to determine how much flour is saved, we subtract the filled volume from the total volume:
[tex]\[ V_{\text{saved}} = V_{\text{total}} - V_{\text{fill}} \][/tex]
[tex]\[ V_{\text{saved}} = 288 \, \text{cubic inches} - 252 \, \text{cubic inches} \][/tex]
[tex]\[ V_{\text{saved}} = 36 \, \text{cubic inches} \][/tex]
### Conclusion:
So the correct answer is not provided in the listed options. We found that the saved volume is 36 cubic inches. There could be a typo or mistake in the options given.
### Step 1: Determine Total Volume of the Box
First, let's calculate the total volume if the box were filled completely. Given the dimensions:
- Height = 12 inches
- Width = 8 inches
- Depth = 3 inches
The volume, [tex]\( V \)[/tex], is calculated by multiplying these dimensions together:
[tex]\[ V_{\text{total}} = \text{height} \times \text{width} \times \text{depth} \][/tex]
[tex]\[ V_{\text{total}} = 12 \, \text{inches} \times 8 \, \text{inches} \times 3 \, \text{inches} \][/tex]
[tex]\[ V_{\text{total}} = 288 \, \text{cubic inches} \][/tex]
### Step 2: Determine Fill Height
Next, the manufacturer wants to leave 1.5 inches unfilled at the top of the box. Thus, the filling height, [tex]\( H_{\text{fill}} \)[/tex], would be:
[tex]\[ H_{\text{fill}} = \text{height} - 1.5 \, \text{inches} \][/tex]
[tex]\[ H_{\text{fill}} = 12 \, \text{inches} - 1.5 \, \text{inches} \][/tex]
[tex]\[ H_{\text{fill}} = 10.5 \, \text{inches} \][/tex]
### Step 3: Calculate the Filled Volume
The volume when the box is filled up to 1.5 inches from the top is calculated similarly, using the fill height:
[tex]\[ V_{\text{fill}} = H_{\text{fill}} \times \text{width} \times \text{depth} \][/tex]
[tex]\[ V_{\text{fill}} = 10.5 \, \text{inches} \times 8 \, \text{inches} \times 3 \, \text{inches} \][/tex]
[tex]\[ V_{\text{fill}} = 10.5 \times 8 \times 3 \][/tex]
[tex]\[ V_{\text{fill}} = 252 \, \text{cubic inches} \][/tex]
### Step 4: Calculate the Saved Volume
Finally, to determine how much flour is saved, we subtract the filled volume from the total volume:
[tex]\[ V_{\text{saved}} = V_{\text{total}} - V_{\text{fill}} \][/tex]
[tex]\[ V_{\text{saved}} = 288 \, \text{cubic inches} - 252 \, \text{cubic inches} \][/tex]
[tex]\[ V_{\text{saved}} = 36 \, \text{cubic inches} \][/tex]
### Conclusion:
So the correct answer is not provided in the listed options. We found that the saved volume is 36 cubic inches. There could be a typo or mistake in the options given.
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