algebra 2 math problems?

Question

Charles has 400 bushels of potatoes that he can now sell for $3.20 per bushel. For every week he waits, the price per bushel will drop by $0.10 but he will harvest 20 more bushels. How many weeks should he wait in order to maximize his income??

math work would be greatly appreciated. i have absolutely no idea where even to start on this problem! thanks!

Answer 1

The assumption is that he can sell everything he has when he decides to do so. pretty unrealistic but we are just doing maths...

The price will be (3.2 - N/10) where N is the number of weeks he waits, and the quantity will be (400+ N 20). So when will the product of the 2 be maximal? One vanishes if N = 32 and the other one if
N = -20. So pick the middle 32 -20 / 2 = 6. That's it!

Answer 2

If the cost to put them to market is cheap enough, he should sell the bushesl as he gets them. Or sell them on e-Bay... haha

Seriously:

The basic formula is this...
He starts with 400 bushels and gets 20 more each week. So, for any given week, he will have how many bushels:
400 + (20 * weeks) note: the asterick symbol * means I multiply.
[e.g. after 3 weeks he will have 400 + (20 * 3) or 460 ... yay! more bushels to sell!]

... for any given week the price per bushel will be:
$3.20 - ( $.10 * Weeks)
[e.g. after 3 weeks the price per bushel will be $3.20 - ($.10 * 3) or $2.90 .... ewww... losing money]

His income will be whatever bushels he has multiplied times the price per bushel. So, we need to multiply the above equations.

Let's let "x" = the number of weeks.
[note: I will use the carrot symbol ^ to represent when we raise a number to a certain power. For example. x times x is the same as x raised to the power of 2. I will write it like: x^2]

His income for any given week will be:

Income =
[#bushels for that week] * [$price per bushel for that week]

(400 + (20 * x)) * (3.20 - (.10 * x))

lose the multiplication symbols so it looks neat...

(400 + (20x)) (3.20 - (.10x))

Multiply the two terms out and we get....

1280 - 40x + 64x - 2x^2

1280 + 24x - 2x^2

- 2x^2 + 24x +1280

So... Income = - 2x^2 + 24x +1280

Let's solve for the peak (or vertex) of the income curve...

"When f(x)=ax^2+bx+c
Vertex coordinates=(-b/2a, f(-b/2a)) " - see reference

So, vertex is where x = -24/2(-2) = -24/-4 = 6

Answer: 6 weeks.

I know there was a lot of typing here. I was hoping to explain, for you, what I was thinking as I typed.

Answer 3

First lets find out how many bushels he has as the weeks go on.

He has 400, but gets 20 more each week he waits, let x be the weeks.

So he has (400+20x) for x weeks.

Now lets find how much he charges per bushel.

He charges 3.20, but lowers the price by .10 per week.

So he charges (3.20-.10x).

To find the price he grosses, multiply the two quantities, we will have number of bushels * $/bushel to leave us with just dollars.

So we have (400+20x)(3.20-.10x). This formula gives us the ammount he makes after x weeks.

To find the max, graph it.

Answer 4

the price of one bushel after x weeks will be 3.2-0.1x

But the number of bushels will be 400+20x

So after x weeks his income

(3.2-0.1x) (400+20x) =1280 +64x-40x -2x^2

the income is after x weeks

-2x^2 +24x +1280 =I

I is maximum for the dI/dx =0

dI/dx= -4x+24

The derivative =0 for x=6 and I =1352

Answer 5

20 weeks

Answer 6

P=3.2
B=400
X=# of weeks

prof=(P-.10X)(B+20X)
=PB-2X^2-.10BX+20PX

20P-.10B-4X=0

X=5P-.10B/4

X=5(3.2)-.10(100)

X=16-10

6 WEEKS BABY!!!!!

Answer 7

ok now its 400x3.20=1280
1st wk is 420x3.10=1302
2nd wk is 440x3.00=1320
3rd wk is 460x2.90=1334
4th wk is 480x2.80=1344
5th wk is 500x2.70=1350
6th wk is 520x2.60=1352
7th wk is 540x2.50=1350 so i would sell in 6th wk @$2.60

Answer 8

holy **** no...