Question
Surface area of bag is 800 square inches. The overhead bin to put them in is 102 by 36 by 33 inches. How many of the bags will fit in the overhead bin? Whole bag is needed….can’t put a .5 bag in
Answers
GPT-4o mini
To determine how many bags will fit in the overhead bin, we can divide the total volume of the bin by the volume of one bag. However, since we only know the surface area of the bag (800 square inches) and not the volume, we'll first need to estimate the dimensions of the bag.
The surface area \( S \) of a bag can be expressed as:
\[
S = 2lw + 2lh + 2wh
\]
where \( l \), \( w \), and \( h \) are the length, width, and height of the bag, respectively.
Given that the surface area is 800 square inches, without loss of generality, we can assume some reasonable dimensions to find the volume.
### Step 1: Calculate the volume of the overhead bin.
The dimensions of the overhead bin are:
- Length = 102 inches
- Width = 36 inches
- Height = 33 inches
Calculating the volume \( V \) of the bin:
\[
V_{\text{bin}} = 102 \times 36 \times 33 = 127,008 \text{ cubic inches}
\]
### Step 2: Estimate possible dimensions for the bag.
To make things easier, we can estimate the dimensions of a bag uniformly. Let's assume the bag has a rectangular shape where width and height are approximately equal.
Let’s say, for example, the bag is a cube with width \( x \).
The formula for the surface area of a cube \( S \) is:
\[
S = 6x^2
\]
Setting this equal to 800 yields:
\[
6x^2 = 800 \implies x^2 = \frac{800}{6} \implies x^2 \approx 133.33 \implies x \approx 11.55 \text{ inches}
\]
Therefore, approximate dimensions of the bag could be \( 12 \times 12 \times 12 \) inches. We can adjust these values for convenience, but let’s go with \( 12 \) inches on each side since it fits reasonably with the surface area condition.
### Step 3: Calculate volume of one bag.
Assuming a bag with dimensions approximately \( 12 \times 12 \times 12 \):
\[
V_{\text{bag}} = 12 \times 12 \times 12 = 1,728 \text{ cubic inches}
\]
### Step 4: Calculate how many bags fit in the bin.
Now, we can find the number of bags that fit into the bin:
\[
\text{Number of bags} = \frac{V_{\text{bin}}}{V_{\text{bag}}} = \frac{127,008}{1,728} \approx 73.5
\]
Since we can’t fit half a bag, we take the whole number:
\[
\text{Number of bags} = 73
\]
### Conclusion
A total of **73 bags** will fit into the overhead bin.
The surface area \( S \) of a bag can be expressed as:
\[
S = 2lw + 2lh + 2wh
\]
where \( l \), \( w \), and \( h \) are the length, width, and height of the bag, respectively.
Given that the surface area is 800 square inches, without loss of generality, we can assume some reasonable dimensions to find the volume.
### Step 1: Calculate the volume of the overhead bin.
The dimensions of the overhead bin are:
- Length = 102 inches
- Width = 36 inches
- Height = 33 inches
Calculating the volume \( V \) of the bin:
\[
V_{\text{bin}} = 102 \times 36 \times 33 = 127,008 \text{ cubic inches}
\]
### Step 2: Estimate possible dimensions for the bag.
To make things easier, we can estimate the dimensions of a bag uniformly. Let's assume the bag has a rectangular shape where width and height are approximately equal.
Let’s say, for example, the bag is a cube with width \( x \).
The formula for the surface area of a cube \( S \) is:
\[
S = 6x^2
\]
Setting this equal to 800 yields:
\[
6x^2 = 800 \implies x^2 = \frac{800}{6} \implies x^2 \approx 133.33 \implies x \approx 11.55 \text{ inches}
\]
Therefore, approximate dimensions of the bag could be \( 12 \times 12 \times 12 \) inches. We can adjust these values for convenience, but let’s go with \( 12 \) inches on each side since it fits reasonably with the surface area condition.
### Step 3: Calculate volume of one bag.
Assuming a bag with dimensions approximately \( 12 \times 12 \times 12 \):
\[
V_{\text{bag}} = 12 \times 12 \times 12 = 1,728 \text{ cubic inches}
\]
### Step 4: Calculate how many bags fit in the bin.
Now, we can find the number of bags that fit into the bin:
\[
\text{Number of bags} = \frac{V_{\text{bin}}}{V_{\text{bag}}} = \frac{127,008}{1,728} \approx 73.5
\]
Since we can’t fit half a bag, we take the whole number:
\[
\text{Number of bags} = 73
\]
### Conclusion
A total of **73 bags** will fit into the overhead bin.