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29.01.2020 •
Mathematics
Aclosed box with a square base is to be built to house an ant colony. the bottom of the box and all four sides are to be made of material costing $1/ft², and the top is to be constructed of glass costing $5/ft². what are the dimensions of the box of greatest volume that can be constructed for $72?
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Ответ:
Dimensions of the box will be 6 ft × 6 ft × 54 feet
Step-by-step explanation:
A closed box is to be built having square base.
Let the dimensions of the box are a ft × a ft × b ft.
Material used to build the base and all four sides cost $1 per square ft.
Therefore, cost of base and four sides = 1[(a)² + 4ab]
Top of the box is to be constructed of glass costing $5 per square feet.
Cost of the top = 5a²
Total cost of the box material = (a² + 4ab) + 5a² = $72
6a² + 4ab = 72
b = -----(1)
Now volume of the box V = a×a×b = a²b
From equation (1),
V = a²()
V = a(18 - 1.5a)
For greatest volume ,
18 - 3a = 0
a = 6
Now form equation (1)
b =
b = 54 feet.
Therefore, dimensions of the box will be 6 ft × 6 ft × 54 feet.
Ответ: