A gas station sells Q gallons of gasoline per year, which is delivered N times per year in equal shipments ofgallons. Thecost of each delivery is d dollars and the yearly storage costs are sQT, where T is the length of time (a fraction of a year)between shipments and s is a constantWhich value of N minimize the costs? Hint: Express T in terms of N.(Use symbolic notation and fractions where needed.)Find the optimal number of deliveries if Q 4 million gal, d = $8000, and s = 31 cents/gal-yr(Your answer should be a whole number, so compare costs for the two integer values of N nearest the optimal value.)N =

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Asked Nov 10, 2019
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A gas station sells Q gallons of gasoline per year, which is delivered N times per year in equal shipments of
gallons. The
cost of each delivery is d dollars and the yearly storage costs are sQT, where T is the length of time (a fraction of a year)
between shipments and s is a constant
Which value of N minimize the costs? Hint: Express T in terms of N.
(Use symbolic notation and fractions where needed.)
Find the optimal number of deliveries if Q 4 million gal, d = $8000, and s = 31 cents/gal-yr
(Your answer should be a whole number, so compare costs for the two integer values of N nearest the optimal value.)
N =
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A gas station sells Q gallons of gasoline per year, which is delivered N times per year in equal shipments of gallons. The cost of each delivery is d dollars and the yearly storage costs are sQT, where T is the length of time (a fraction of a year) between shipments and s is a constant Which value of N minimize the costs? Hint: Express T in terms of N. (Use symbolic notation and fractions where needed.) Find the optimal number of deliveries if Q 4 million gal, d = $8000, and s = 31 cents/gal-yr (Your answer should be a whole number, so compare costs for the two integer values of N nearest the optimal value.) N =

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Expert Answer

Step 1

Let N be the shipment per year and the time interval between the shipments is T= 1/N.

The storage cost per year is sQ/N and yearly delivery cost is dN.

That is, the total cost functionis C(N)=dN +-
sQ
Differentiate C(N)=dN +
with respect to N
N
d
C'(N)
dN
=N-
N2
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That is, the total cost functionis C(N)=dN +- sQ Differentiate C(N)=dN + with respect to N N d C'(N) dN =N- N2

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Step 2
Equate the derivative to zero to find the value ofN
sQ
N
N2
sQ
N
d
Thus, the value of N =
d
sQ
Substitute Q 4000000,d $8000 and s 0.31 in N=,
0.31x4,000,000
N =
8000
=12.4499
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Equate the derivative to zero to find the value ofN sQ N N2 sQ N d Thus, the value of N = d sQ Substitute Q 4000000,d $8000 and s 0.31 in N=, 0.31x4,000,000 N = 8000 =12.4499

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