the number of rectangular containers to maximze their income is 18 while the number of cylindrical containers to maximize income is 12.
The maximum income is $1740
Let x be the number of rectangular containers and y the number of cylindrical containers.
Since a rectangular container has a volume of 100 ft ³and weighs 200 pounds and a cylindrical container has a volume of 200 ft.³ and weighs 100 pounds, and each truck has room for at most 4200 ft.³ of containers and can carry a maximum of 4800 pounds,
We have that the maximum volume of the truck V = 100x + 200y.
Also, the maximum weight of the truck is W = 200x + 100y
Since V = 4200 ft.³ and W = 4800 pounds,
100x + 200y = 4200 and 200x + 100y = 4800
x + 2y = 42 (1) and 2x + y = 48 (2)
Multiplying (1) by 2 and (2) by 1, we have
2x + 4y = 84 (3) and 2x + y = 48 (4)
Subtracting (4) from (3), we have
2x + 4y = 84
-
2x + y = 48
3y = 36
y = 36/3
y = 12
Substituting y into (1), we have
x + 2y = 42
x + 2(12) = 42
x + 24 = 42
x = 42 - 24
x = 18
Also, since the shipping company charges $60 for a rectangular container and $55 for a cylindrical container, the income, P = 60x + 55y
Since x = 18 and y = 12, the maximum income is P = 60x + 55y
= 60(18) + 55(12)
= 1080 + 660
= $1740
So, the number of rectangular containers to maximze their income is 18 while the number of cylindrical containers to maximize income is 12.
The maximum income is $1740
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