Posted by Frank on Tuesday, February 14, 2012 at 5:33pm.
Open boxes are being designed to hold mail for the post office. The boxes will be made to have the greatest possible volume. Each box will be made from a sheet of metal that measures 15 inches by 12 inches. They will be formed by cutting and removing a square from each corner. The sides will then be folded up and welded together along each corner. What is the maximum volume?
- math - Reiny, Tuesday, February 14, 2012 at 7:18pm
Let each side of the square to be cut out be x inches
length of base = 15-2x
width of base = 12-2x
height of box = x
where 0 < x < 6 or else the dimensions make no sense
Volume = x(15-2x)(12-2x)
= 180x - 54x^2 + 4x^3
d(Volume)/dx = 180 - 108x + 12x^2
= 0 for a max/min of Volume
12x^2 - 108x + 180 = 0
x^2 - 9x + 15 = 0
x = (9 ± √21)/2
x = 2.2087 or x = 20.62 , but x < 6
so x = 2.2087
Maximum Volume = 2.2087(10.5826)(7.5826) = 177.2340895 cubic inches
let x = 2.2
Volume = 2.2(10.6)(7.6) = 177.232 , a bit smaller
let x = 2.21
Volume = 2.21(10.58)(7.58) = 177.234044 a tiny bit smaller
Answer This Question
More Related Questions
- Math - A mail truck picks up two boxes of mail from the post office. The total ...
- calculus - A candy company needs a custom box for their truffles. The box they'...
- 5th grade math (word problems) - Maria is packaging boxes into a carton that is ...
- Calculus - A carpenter wants to make an open-topped box out of a rectangular ...
- Math - A machine produces open boxes using rectangular sheets of metal (12in by ...
- ALGEBRA - DAmon you equaled them to zero that is not rightt An open-topped box ...
- Calculus - I actually have two questions: 4. An open box is to be made from a ...
- Maths - Algebra - Boxes are made by cutting 8cm squares from the corners of ...
- calculus - an open box is to be made from a piece of metal 16 by 30 inches by ...
- Geometry - A square piece of cardboard, 24 inches by 24 inches, is to be made ...