Question
Surface area of bag is 800 sq inches. The overhead bin to put them in is 102x36x33 inches.
How many of the bags will fit in the overhead bin? Whole bag is needed..can't put a .5 bag in.
How many of the bags will fit in the overhead bin? Whole bag is needed..can't put a .5 bag in.
Answers
Answered by
GPT-4o mini
To determine how many bags can fit in the overhead bin, we need to calculate the volume of both the bags and the overhead bin.
1. **Calculate the volume of the overhead bin:**
The dimensions of the bin are:
- Length = 102 inches
- Width = 36 inches
- Height = 33 inches
Volume of the bin \( V_{bin} = \text{Length} \times \text{Width} \times \text{Height} \)
\[
V_{bin} = 102 \text{ in} \times 36 \text{ in} \times 33 \text{ in} = 122,496 \text{ cubic inches}
\]
2. **Calculate the volume of one bag:**
Given that the surface area of one bag is 800 sq inches, we cannot directly calculate its volume without knowing its shape. For simplicity, let's assume the bag is a rectangular prism and try to estimate its dimensions based on the surface area.
The surface area \( SA \) of a rectangular prism is given by:
\[
SA = 2(lb + lh + wh)
\]
For our purposes, we will work with the known surface area and make some assumptions or estimations to determine an approximate volume.
A common assumption for the dimensions could be:
- Length \( l = x \)
- Width \( w = y \)
- Height \( h = z \)
Therefore, we can try a few combinations or recognize that without specific information, it's difficult to provide an exact volume.
However, if we were to approximate the volume for a bag based on the surface area being \( SA = 800 \) sq in, let’s pick an arbitrary set of dimensions to estimate its volume. For simplicity, assume the bag is somewhat cuboidal with equal dimensions.
If we assume a cubic approximation:
- Let \( x \) be the side length.
- So, \( SA = 6x^2 = 800 \)
- Thus, \( x^2 = \frac{800}{6} \approx 133.33 \) and \( x \approx 11.55 \) (roughly) inches.
- The volume of one bag then is:
\[
V_{bag} = x^3 \approx (11.55)^3 \approx 1,540.92 \text{ cubic inches}
\]
For rigorous packing, this may not be exact, but lets proceed with this estimate.
3. **Calculate how many bags fit:**
Now we can calculate how many bags can fit into the overhead bin by dividing the volume of the bin by the volume of one bag:
\[
\text{Number of bags} = \frac{V_{bin}}{V_{bag}} = \frac{122496 \text{ cu in}}{1540.92 \text{ cu in}} \approx 79.5
\]
Since we cannot fit half a bag, we round down to the nearest whole number.
Therefore, **the maximum number of whole bags that can fit in the overhead bin is 79**.
1. **Calculate the volume of the overhead bin:**
The dimensions of the bin are:
- Length = 102 inches
- Width = 36 inches
- Height = 33 inches
Volume of the bin \( V_{bin} = \text{Length} \times \text{Width} \times \text{Height} \)
\[
V_{bin} = 102 \text{ in} \times 36 \text{ in} \times 33 \text{ in} = 122,496 \text{ cubic inches}
\]
2. **Calculate the volume of one bag:**
Given that the surface area of one bag is 800 sq inches, we cannot directly calculate its volume without knowing its shape. For simplicity, let's assume the bag is a rectangular prism and try to estimate its dimensions based on the surface area.
The surface area \( SA \) of a rectangular prism is given by:
\[
SA = 2(lb + lh + wh)
\]
For our purposes, we will work with the known surface area and make some assumptions or estimations to determine an approximate volume.
A common assumption for the dimensions could be:
- Length \( l = x \)
- Width \( w = y \)
- Height \( h = z \)
Therefore, we can try a few combinations or recognize that without specific information, it's difficult to provide an exact volume.
However, if we were to approximate the volume for a bag based on the surface area being \( SA = 800 \) sq in, let’s pick an arbitrary set of dimensions to estimate its volume. For simplicity, assume the bag is somewhat cuboidal with equal dimensions.
If we assume a cubic approximation:
- Let \( x \) be the side length.
- So, \( SA = 6x^2 = 800 \)
- Thus, \( x^2 = \frac{800}{6} \approx 133.33 \) and \( x \approx 11.55 \) (roughly) inches.
- The volume of one bag then is:
\[
V_{bag} = x^3 \approx (11.55)^3 \approx 1,540.92 \text{ cubic inches}
\]
For rigorous packing, this may not be exact, but lets proceed with this estimate.
3. **Calculate how many bags fit:**
Now we can calculate how many bags can fit into the overhead bin by dividing the volume of the bin by the volume of one bag:
\[
\text{Number of bags} = \frac{V_{bin}}{V_{bag}} = \frac{122496 \text{ cu in}}{1540.92 \text{ cu in}} \approx 79.5
\]
Since we cannot fit half a bag, we round down to the nearest whole number.
Therefore, **the maximum number of whole bags that can fit in the overhead bin is 79**.