otter pop frenzy
TRANSCRIPT
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Otter Pop Frenzy
By Karly Kelso
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Costco is revamping their computer programs, but their
system is currently shut down.
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They need to negotiate a new contract with the makers of Otter Pops immediately, and they only have a few random graphs with
equations from a previous presentation to look at.
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If Otter Pops were sold year round, the amount of people buying them
can be shown as
y(t)=-2.413x³+24.715x²+86.307x-227.867where t is the month and t=1 is January
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How many Otter Pops could potentially be sold?
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ANSWER
The way to figure out this answer is to look at the area
under the curve or the integral of the equation.
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If you take the integral of y(t)=-2.413x³+24.715x²+86.307x-227.867
You get:-.603x4+8.238x³+43.154x²-227.867x│ from 1 to 12
plugging in Y(12)-Y(1) gives the answer of approximately 5384 boxes of Otter Pops
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You can also plug this equation into your calculator and graph it and then hit 2nd Calc and 7 to calculate the
integral from 1 to 12.
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If sold at $9.49 per box (which happens to be the highest price they can be sold at), how much
money would be made if the total potential sold could be
reached?
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ANSWER
This problem is a little more simple. You simply multiply the number you reached in
part A by 9.49 to get the answer $51,094.16
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HOWEVER, the Otter Pop company will sell Otter Pops at a
lower price when the quantity bought is higher, but the boxes can
only be purchased during the month that Costco is planning on
selling them…
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The price of each Otter Pop when bought in a bulk is modeled by the equation
y(b)=21.525-1.978lnbwhere b is the total number
purchased and y(b) is the price per Otter Pop box.
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If 10% profit must be made per box, when can Costco afford to purchase these Otter Pops from
the company?
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ANSWER
First you must calculate the highest price that the Otter
Pops can be bought at to earn a profit.
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This amount will be equal to 9.49/1.1 because the general equation is 110% X the price
bought=the price sold at.So the price bought=the price
sold at/110%(or 1.1)This equals $8.63
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Now the next trick to solving the equation is to set $8.63 equal to
the equation y(b)=21.525-1.978lnb
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8.63=21.525-1.978lnb-12.895=-1.978lnb
6.519=lnbe6.519=b678=b
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This value of b says that Otter Pops must be purchased in
orders larger than 678 in order to earn enough profit. Now you must set THIS number equal toy(t)=-2.413x³+24.715x²+86.307x-227.867
to find out which months are profitable
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It is profitable when the equation is greater than 678 so you set this equation equal to
678 and solve for x.
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However, an easier way to solve the problem is to graph
y(t)=-2.413x³+24.715x²+86.307x-227.867 and y=678
and use your calculator to find the intersection points of these
two graphs
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When you do this, you find that the intersection points are
AROUND 6 and 10, meaning that Otter Pops should be sold
between June and Octoberin order to make a high enough
profit
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The Otter Pop company also wanted to know for which month they should expect the largest
order from Costco
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For this, you must find the derivative of the original
equation
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Original equation y(t)=-2.413x³+24.715x²+86.307x-227.867
Derivative of equationy’(t)=-7.239x²+49.43x+86.307
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The month for the largest amount of Otter Pops sold is a
maximum. To find the maximum, check the endpoints,
where the derivative equals zero, and where the derivative
does not exist.
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To find where the derivative equals zero, you graph the
equation and hit 2nd calc and 2. This produces an answer of 8.3.
Also, the derivative exists everywhere.
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So now you must check where x=1,8.3, and 12
x=1, y=-119.3x=8.3, y=811.38x=12, y=197.11
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Therefore, 8.3 is a maximum so August is the month where the most number of Otter Pops are
sold.