High School

A restaurant has an annual demand for 915 bottles of California wine. It costs $1 to store 1 bottle for 1 year, and it costs $8 to place a reorder.

(a) Find the optimum number of bottles per order.
(b) How many times a year should the wine be ordered?

(a) The optimum number of bottles per order is _____. (Type a whole number.)
(b) The wine should be ordered _____ times a year. (Round to one decimal place as needed.)

Answer :

Therefore, the wine should be ordered 11 times a year (rounded to the nearest integer).

(a) The optimum number of bottles per order is 228. Answer more than 100 words: Let x be the number of bottles per order, then there will be (915/x) orders per year.

So, the total cost of a year would be $915 (for storage) and (915/x)(8) dollars (for orders). Therefore, the cost function C(x) is given by:

C(x)=915+(7320/x)

To find the optimum number of bottles per order we need to differentiate C(x) and equate it to zero:∂C(x)/∂x=−7320/x²

=0⇒x²

=7320⇒x

=√(7320)≈85.5

Therefore, the optimum number of bottles per order is 86 bottles.

Note that the second derivative of C(x) is positive for x > 0, so C(x) is convex upwards and, in particular, C(86) is the minimum value. So the minimum cost is: C(86)≈$1108.72

(b) The wine should be ordered (11) times a year.

To find how many times a year the wine should be ordered, we substitute x = 86 into the formula of the number of orders per year:

915/x=915/86≈10.64

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