5. Agricultural research on a certain variety of fruit has...

5. Agricultural research on a certain variety of fruit has shown that if 30 trees are planted per acre, then each tree will yield an average of 300 pounds of fruit per season. For each additional tree planted per acre, the average yield per tree is reduced by 5 pounds. a. Find an equation for the yield, Y, of a 1-acre plot as a function of the number of trees planted. b. How many trees should be planted per acre to produce the maximum yield? What is the maximum yield per acre?

Transcript

00:01 So in this problem, we're called that the agricultural research on a certain variety of fruit has shown that if 30 trees are planted per acre, then each tree is going to yield an average of 300 pounds of fruit per season.

00:13 So again, that's per tree.

00:14 But then we're told for each additional tree that's planted per acre, the average yield per tree is actually going to go down by five pounds.

00:21 So in part a, we want to figure out an equation for the yield.

00:25 So our equation is going to be in terms of why of a one acre plot as a function of the number of trees planted.

00:30 All right, well, remember, initially, if we had 30 trees, each of them would produce 300 pounds, meaning we would take the number of trees and we would multiply it by 300 pounds to get the total.

00:42 Well, in this take, they say for every time we add a tree, but we don't know how many trees we're going to add, so we're going to call that x.

00:49 So 30 plus x would represent the number of trees.

00:52 Now we're told that the yield for every time they go up by a tree, the yield's going to go down by five pounds.

00:57 Well, initially it's starting at 300 per tree, but it's going down 5 for every additional tree.

01:03 So it subtract this by 5x.

01:05 Perfect.

01:06 Now we have our function.

01:08 Now, in part b, it says how many trees should be planted per acre to produce the maximum yield? and then what is the maximum yield? well, as you can see here, we can go ahead and multiply out this function.

01:20 So we're going to use the foil method.

01:22 So we're going to have 30 times 300, which is equal to 9 ,000.

01:26 Then we have 30 times negative 5x, which would be negative 150x.

01:31 Next, x times 300 would be positive 300x...

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