precalculus question

Question

in a single growing season at the Smith Family Orchard, the average yield per apple tree is 150 apples when the number of trees per acre is 100. For each additional tree over 100, the average yield per tree decreases by 1.

What would be the average yield per tree if the number of trees per acre was doubled? What would be the total yield in that case?

How many trees should be planted per acre to maximize the total yield?

What type of function did you use to model the scenario? Be specific.

Why did you choose that type of function? What characteristics of the model led you to make that choice?

Did you write a formula for the function? If not, do so now. Be sure to state what the independent and dependent variable represent in the model.

Answer 1

Hi Monica,

Here's the most difficult part done for you.

n=# of new trees planted and also # of apples less each tree produces

(100+n)(150-n)=0

15,000+150n-100n-n^2=0

15,000+50n-n^2=0

this is a parabola that opens downward so there is a maximum at the vertex, (h,k)

h=-b/2a is the maximizing point

h=-50/-2

h=25

you can substitute 25 for n in the equation and get the maximum or substitute into ((100+n)(150-n)

to get the maximum

(100+25)(150-25)=125*125=15,625 is the maximum number of apples

if you planted 24 tees you would get (100+24)(150-24)=124*126=15,624 which is less apples by one

also if you planted 26 trees you get 126*124=15,624 again

if you double the number of trees you get (100+100)(150-100)=200*50=10,000 apples