the number of shipments each week from the diamond nines of Arkansas to a jewelr
ID: 2892056 • Letter: T
Question
the number of shipments each week from the diamond nines of Arkansas to a jewelry wholesaler is expressed by the equation shipment in week x = 35 – 1.8x + .04x^2 where x is the number of weeks after January 1st. Meanwhile, sales of diamonds each week is sales in week x = 15 + .1x where x is the number of weeks after January 1st
a. If inventory of diamonds was 0 on January 1st, how much unsold inventory would remain after 52 weeks? Keep in mind that inventory = total number of shipments – total sales to date
b. Over the range 0 *less than or equal to* x *less than or equal to* 52, during which week was the number of shipments the lowest?
c. During the weeks when shipments exceeded sales, how many more diamonds were shipped than were sold?
d. What was the average level of inventory over the 52 weeks?
Explanation / Answer
a> inventory = total number of shipments – total sales to date
after 52 weeks
inventory = 52[(35 – 1.8x + .04x^2) - (15 + .1x)]
inventory ,I = 52[20 - 1.9x + .04x^2] , I(0)=0
b> Shipments , S(x) = (35 – 1.8x + .04x^2)
S'(x) = -1.8 + .08x
S'(x) = -1.8 + .08x = 0
x = 22.5
S(22.5) = (35 – 1.8(22.5) + .04(22.5)^2) = 14.75
as S(22.5) > 0
so we get a minima at x = 22.5
hence in the 22.5 or 23 rd week (approximately) the number of shipments was the lowest
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