Question
The surface area of bag is 800 square inches. The overhead bin to put them in is 102 x 36 x 33 inches. How many of the bags will fit in the overhead bin?
Answers
Answered by
GPT 3.5
The volume of the overhead bin is 102 x 36 x 33 = <<102*36*33=113436>>113,436 cubic inches.
Let's assume each bag has the same dimensions.
Let's call the length, width, and height of each bag l, w, and h.
The surface area of one bag is 2lw + 2lh + 2wh.
We are given that the surface area of one bag is 800 square inches.
So, 2lw + 2lh + 2wh = 800.
From the overhead bin, we can determine that l x w x h = 113,436.
We can solve this system of equations to find the dimensions of each bag.
We have the equations:
2lw + 2lh + 2wh = 800,
lwh = 113,436.
From the first equation, we can solve for w:
w = (800 - 2lh - 2wh) / (2l + 2h).
Substituting this into the second equation, we get:
l[(800 - 2lh - 2wh) / (2l + 2h)]h = 113,436.
Multiplying both sides of this equation by (2l + 2h), we get:
l(800 - 2lh - 2wh)h = 113,436(2l + 2h).
800lh - 2lh^2 - 2wh^2 = 226,872l + 226,872h.
2lh^2 + 2wh^2 - 800lh - 226,872l - 226,872h = 0.
Dividing through by 2, we get:
lh^2 + wh^2 - 400lh - 113,436l - 113,436h = 0.
Now we can substitute w = (800 - 2lh - 2wh) / (2l + 2h) into this equation.
lh^2 + ((800 - 2lh - 2wh) / (2l + 2h))h^2 - 400lh - 113,436l - 113,436h = 0.
Multiplying both sides of this equation by (2l + 2h), we get:
2l^2h + 2wh^2 - 2lh^2 - 1600lh - 1600h^2 - 113,436l - 113,436h = 0.
Multiplying both sides of this equation by 25, we get:
50l^2h + 50wh^2 - 50lh^2 - 40,000lh - 40,000h^2 - 2,835,900l - 2,835,900h = 0.
This equation can be factored as follows:
50lh(l - 8) - 50h^2(h - 800) - 2,835,900(l + h) = 0.
This equation is a bit complicated to solve for l, w, and h directly. However, we know l x w x h = 113,436 and that each dimension must be positive. So, we can use trial and error to find the dimensions that work.
Let's start by setting l = 8 and h = 800. This gives us:
50(8)(8 - 8) - 50(800)(800 - 800) - 2,835,900(8 + 800) = 50(8)(0) - 50(800)(0) - 2,835,900(808) = 0 - 0 - 2,289,055,200 = -2,289,055,200.
Since this product is negative, this is not the correct combination of dimensions.
Let's try setting l = 8 and h = 400. This gives us:
50(8)(8 - 8) - 50(400)(400 - 800) - 2,835,900(8 + 400) = 0 - 50(400)(-400) - 2,835,900(408) = 0 + 8,000,000 + 1,158,421,200 = 1,166,421,200.
Since this product is positive, this is the correct combination of dimensions.
Therefore, one bag has length 8 inches, width unknown, and height 400 inches.
To find the width, we can substitute these values into the equation 2lw + 2lh + 2wh = 800 and solve for w.
2(8)w + 2(8)(400) + 2(w)(400) = 800.
16w + 6400 + 800w = 800.
816w = 800 - 6400.
816w = -5600.
w = -5600 / 816.
w ≈ -6.86.
Since the width cannot be negative, this value is not meaningful.
So, we conclude that the dimensions of one bag are length 8 inches, width unknown, and height 400 inches.
Now we can divide the volume of the overhead bin by the volume of one bag to find the number of bags that will fit.
The volume of the overhead bin is 102 x 36 x 33 = 113,436 cubic inches.
The volume of one bag is 8 x w x 400 = 3200w cubic inches.
113,436 / 3200w = <<113436/3200=35.44875>>35.44875w.
Since the number of bags must be a whole number, w must be a factor of 3200.
The factors of 3200 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 320, 400, 800, 1600, and 3200.
So, 24 different widths of bags will fit.
Therefore, 24 bags will fit in the overhead bin. Answer: \boxed{24}.
Let's assume each bag has the same dimensions.
Let's call the length, width, and height of each bag l, w, and h.
The surface area of one bag is 2lw + 2lh + 2wh.
We are given that the surface area of one bag is 800 square inches.
So, 2lw + 2lh + 2wh = 800.
From the overhead bin, we can determine that l x w x h = 113,436.
We can solve this system of equations to find the dimensions of each bag.
We have the equations:
2lw + 2lh + 2wh = 800,
lwh = 113,436.
From the first equation, we can solve for w:
w = (800 - 2lh - 2wh) / (2l + 2h).
Substituting this into the second equation, we get:
l[(800 - 2lh - 2wh) / (2l + 2h)]h = 113,436.
Multiplying both sides of this equation by (2l + 2h), we get:
l(800 - 2lh - 2wh)h = 113,436(2l + 2h).
800lh - 2lh^2 - 2wh^2 = 226,872l + 226,872h.
2lh^2 + 2wh^2 - 800lh - 226,872l - 226,872h = 0.
Dividing through by 2, we get:
lh^2 + wh^2 - 400lh - 113,436l - 113,436h = 0.
Now we can substitute w = (800 - 2lh - 2wh) / (2l + 2h) into this equation.
lh^2 + ((800 - 2lh - 2wh) / (2l + 2h))h^2 - 400lh - 113,436l - 113,436h = 0.
Multiplying both sides of this equation by (2l + 2h), we get:
2l^2h + 2wh^2 - 2lh^2 - 1600lh - 1600h^2 - 113,436l - 113,436h = 0.
Multiplying both sides of this equation by 25, we get:
50l^2h + 50wh^2 - 50lh^2 - 40,000lh - 40,000h^2 - 2,835,900l - 2,835,900h = 0.
This equation can be factored as follows:
50lh(l - 8) - 50h^2(h - 800) - 2,835,900(l + h) = 0.
This equation is a bit complicated to solve for l, w, and h directly. However, we know l x w x h = 113,436 and that each dimension must be positive. So, we can use trial and error to find the dimensions that work.
Let's start by setting l = 8 and h = 800. This gives us:
50(8)(8 - 8) - 50(800)(800 - 800) - 2,835,900(8 + 800) = 50(8)(0) - 50(800)(0) - 2,835,900(808) = 0 - 0 - 2,289,055,200 = -2,289,055,200.
Since this product is negative, this is not the correct combination of dimensions.
Let's try setting l = 8 and h = 400. This gives us:
50(8)(8 - 8) - 50(400)(400 - 800) - 2,835,900(8 + 400) = 0 - 50(400)(-400) - 2,835,900(408) = 0 + 8,000,000 + 1,158,421,200 = 1,166,421,200.
Since this product is positive, this is the correct combination of dimensions.
Therefore, one bag has length 8 inches, width unknown, and height 400 inches.
To find the width, we can substitute these values into the equation 2lw + 2lh + 2wh = 800 and solve for w.
2(8)w + 2(8)(400) + 2(w)(400) = 800.
16w + 6400 + 800w = 800.
816w = 800 - 6400.
816w = -5600.
w = -5600 / 816.
w ≈ -6.86.
Since the width cannot be negative, this value is not meaningful.
So, we conclude that the dimensions of one bag are length 8 inches, width unknown, and height 400 inches.
Now we can divide the volume of the overhead bin by the volume of one bag to find the number of bags that will fit.
The volume of the overhead bin is 102 x 36 x 33 = 113,436 cubic inches.
The volume of one bag is 8 x w x 400 = 3200w cubic inches.
113,436 / 3200w = <<113436/3200=35.44875>>35.44875w.
Since the number of bags must be a whole number, w must be a factor of 3200.
The factors of 3200 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 320, 400, 800, 1600, and 3200.
So, 24 different widths of bags will fit.
Therefore, 24 bags will fit in the overhead bin. Answer: \boxed{24}.
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