Max/min word prob.: Farmers can get $2/bushel, but then....

[startingover]

Farmers can get $2 per bushel for their potatoes on July 1st, and after that, the price drops by 2 cents per bushel per day. On July 1st, a farmer has 80 bushels of potatoes in the field and estimates that the crop is increasing at the rate of 1 bushel per day. When should the farmer harvest the potatoes to maximize his profit.

So, I should let x equal the number of days after July 1st.

I'm thinking the problem would be
R = (80 bushels + 1x)(x - .2)

 

[soroban]

Re: Word Problem Set up

Hello, startingover!

Farmers can get $2 per bushel for their potatoes on July 1st,
and after that, the price drops by 2 cents per bushel per day.
On July 1st, a farmer has 80 bushels of potatoes in the field
and estimates that the crop is increasing at the rate of 1 bushel per day.
When should the farmer harvest the potatoes to maximize his profit.

So, I should let \(\displaystyle x\) equal the number of days after July 1st.

I'm thinking the problem would be:
. . R = (80 bushels + 1x)(x - .2) . . . . no

What is \(\displaystyle (x\,-\,0.2)\) ? . . . "Number of days" minus "twenty cents" ?

The first factor is correct.
. . On day \(\displaystyle x\), the number of bushels is: \(\displaystyle \,80\,+\,1x\)

The second factor is the price per bushel.
. . On day \(\displaystyle x\), the price is: \(\displaystyle \,(2\,-\,0.02x)\)

Therefore: \(\displaystyle \L\:R \:=\80\,+\,x)(2\,-\,0.02x)\)

 

[tkhunny]

Re: Word Problem Set up

R = (80 bushels + 1x)(x - .2)

You are having some units problems. You should fix those, particularly making up your mind if you are going to use units or not.

Other than that, you are REALLY close:
0.2 = 20¢
0.02 = 2¢

 

[soroban]

Hello, starting over!

For some reason, it's not letting me post my reply.

\(\displaystyle R(x) \:=\: (80\,+\,x)(2\,-\,0.02x) \:=\:160\,-\,1.6x\,+\,2x\,-\,0.02x^2 \:=\:-0.02x^2\,+\,0.4x\,+\,160\)

\(\displaystyle R'(x) \:=\:-0.04x\,+\,0.4\:=\:0\;\;\Rightarrow\;\;0.04x\:=\:0.4\;\;\Rightarrow\;\;x\,=\,10\)

So 10 days after July 1st, which would be July 11.

Absolutely correct! . . . Nice work!