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Answer :
The optimum number of bottles per order is approximately 44 bottles, and the wine should be ordered 21 times per year.
To find the optimum number of bottles per order, we need to minimize the total cost of storage and reordering.
Let's assume the number of bottles per order is x.
(a) To minimize the total cost, we need to consider the cost of storage and the cost of placing reorders.
The cost of storage is given by 3 dollars per bottle per year. Therefore, the total cost of storage per year is 3x.
The cost of placing a reorder is a fixed cost of 8 dollars per order. To determine the number of orders per year, we divide the annual demand by the number of bottles per order: 904/x.
The total cost per year is the sum of the cost of storage and the cost of placing reorders:
Total cost = Cost of storage + Cost of reordering
= 3x + 8(904/x)
To find the optimum number of bottles per order, we can find the minimum of the total cost by differentiating the cost function with respect to x and setting it equal to zero:
d/dx (3x + 8(904/x)) = 0
Differentiating, we get:
[tex]3 - 8(904/x^2) = 0[/tex]
Simplifying:
3x^2 = 8(904)
x^2 = 8(904)/3
x^2 ≈ 1921.33
x ≈ √1921.33
x ≈ 43.85
Therefore, the optimum number of bottles per order is approximately 44 bottles.
(b) To determine the number of times the wine should be ordered per year, we divide the annual demand by the number of bottles per order:
Number of orders per year = 904/44 ≈ 20.55
Since we cannot have a fraction of an order, we round up to the nearest whole number. Therefore, the wine should be ordered 21 times per year.
In summary, the optimum number of bottles per order is approximately 44 bottles, and the wine should be ordered 21 times per year.
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A restaurant has an annual demand for904bottles of California wine. It costs$3to store 1 bottle for 1 year, and it costs$8to place a reorder.(a) Find the optimum number of bottles per order.(b) How many times a year should the wine be ordered?
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