Otter Pop Frenzy
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Transcript of Otter Pop Frenzy
Otter Pop Frenzy
By Karly Kelso
Costco is revamping their computer programs, but their
system is currently shut down.
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.
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
How many Otter Pops could potentially be sold?
ANSWER
The way to figure out this answer is to look at the area
under the curve or the integral of the equation.
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
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.
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?
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
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…
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.
If 10% profit must be made per box, when can Costco afford to purchase these Otter Pops from
the company?
ANSWER
First you must calculate the highest price that the Otter
Pops can be bought at to earn a profit.
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
Now the next trick to solving the equation is to set $8.63 equal to
the equation y(b)=21.525-1.978lnb
8.63=21.525-1.978lnb-12.895=-1.978lnb
6.519=lnbe6.519=b678=b
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
It is profitable when the equation is greater than 678 so you set this equation equal to
678 and solve for x.
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
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
The Otter Pop company also wanted to know for which month they should expect the largest
order from Costco
For this, you must find the derivative of the original
equation
Original equation y(t)=-2.413x³+24.715x²+86.307x-227.867
Derivative of equationy’(t)=-7.239x²+49.43x+86.307
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.
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.
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
Therefore, 8.3 is a maximum so August is the month where the most number of Otter Pops are
sold.