00:01
A farmer that raises strawberries will receive $28 per bushel of strawberries during the first week of harvesting.
00:06
The value of strawberries will drop each of the following weeks at 60 cents per bushel, and there are currently 119 bushels of strawberries in the fields, and the crop is increasing at a rate of four bushels per week.
00:21
So when should the farmer harvest the strawberries to maximize his revenue? we're going to let week one be the first week of harvesting.
00:30
If we were to create a little table here's a week number and here's the price that he will receive per bushel at week one he's getting 28 this is going to be price per bushel here so it's price per bushel at week one is 28 and at week one currently there are 119 bushels so this will be the quantity quantity at 119 bushels so right now if we were to figure out the total revenue since he's wanting to maximize his revenue how much money would he get well at 28 bushels or $28 per barrel that's in dollars and he has 119 bushels this here is in bushels we do 28 times 119 that gives us 3 ,332 dollars so that's his revenue right now what about at week two well it's gonna drop his price will drop 60 cents as it told us that would go to 2740 but his quantities increases.
02:03
He's going to get four bushels per week, so he goes up to 123.
02:06
He gets four more bushels.
02:08
If we multiply those together, 123 bushels at 2740 per bushel.
02:13
If we multiply those, that gives us 3 ,370 and 20 cents.
02:18
So it looks like he can make more money.
02:21
If we do the third week, the price would drop 60 more cents.
02:26
That would go to 2680, but he gets four more bushels.
02:30
So that would go up to 127.
02:32
If we multiply those together, we get 3 ,403 .60.
02:41
So we can see that the price is going down.
02:45
The quantity is going up.
02:46
So far, it looks like the revenue is going up, but since eventually the price will drop to zero.
02:52
We know that.
02:53
And then it doesn't matter how many bushels he has, he's not going to make any revenue.
02:57
So this revenue is going to max out at some point.
02:59
So if we let this be week x, week x, the price per bushel is dropping 60 cents, so that we're going to say negative 0 .6 times x.
03:13
And then we need to add our initial value.
03:16
What would it be at week zero since week one is starting at 28? well, week zero would be up just 60 cents from 28, so that would be 28.
03:27
Okay, and what about the bushels, the quantity? well, that depends on the rate which is going up for per week.
03:36
So this would be four times however many weeks has passed, plus our initial value at week zero at week one since he had 119.
03:45
That means at week zero, he would have four less, though it would be 115.
03:48
So our revenue would be these two expressions multiply together.
03:52
So that would be negative 0 .6x plus 2860 times 4x plus 1 .6 .5 .5 .5.
04:02
And we can multiply those out using the distributive property multiply these x terms together 4x times negative 0 .6x that would give us negative 2 .4x squared we multiply negative 0 .6x times 115 that gives negative 69x and then 2860 times 4x that would give us plus 114 .4x, and then 2860 times 115, that would give us plus 3 ,289.
04:45
So that shows us our revenue.
04:49
We can combine these like terms here that gives us negative 2 .4x squared, and if we combine those, that gives us plus 45 .4x, and we had the plus 3 ,289.
05:06
So we're trying to figure out when this would be maximized.
05:09
Well, this is a quadratic function.
05:11
We know it's quadratic and that means it would be a parabola and it opens down since this leading coefficient is negative...