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Narutokun
12.11.2019 •
Mathematics
Astorage shed is to be built in the shape of a box with a square base. it is to have a volume of 686 cubic feet. the concrete for the base costs $8 per square foot, the material for the roof costs $6 per square foot, and the material for the sides costs $3.50 per square foot. find the dimensions of the most economical shed.
the length of the's base is x ft
the height of the shed is x ft
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Ответ:
(8.44 ft. × 8.44 ft × 9.63 ft.)
Step-by-step explanation:
A storage shed has the volume = 686 cubic feet.
686 = x²h ----------------(1)
Let the length of square base = x ft
and height of the shed = h ft
Area of concrete base = x² ft²
Cost to build the base = 8x² [since cost to build the square base = $8 per square feet]
Area of the sides of the shed = 4xh
Cost to build the sides = 4xh (3.50) [since cost to build the sides = $3.50 per square ft]
C = 14xh
Total cost = 8x² + 14xh
(C) = 8x² + (14x) ×
= 8x² + ![(\frac{9604}{x})](/tpl/images/0370/1511/842b8.png)
To minimize the cost we will take the derivative of c and equate it to 0.
16x³ = 9604
x³ =![\frac{9604}{16}](/tpl/images/0370/1511/9d84c.png)
x³ = 600.25
x = 8.44 ft.
For x = 8.44
686 = (8.44)² h
h =![\frac{686}{(8.44)^2}](/tpl/images/0370/1511/7d76e.png)
= 9.63 ft
Therefore, for the minimum cost, dimension of the shed should be (8.44 ft. × 8.44 ft × 9.63 ft.)
Ответ: