3. 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.
a. 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?
b. how many trees should be planted per acre to maximize the total yield?

A.10000

B.25 more trees must be planted

Step-by-step explanation:

⇒Given:

A.

⇒If the total trees per acre is doubled , which means :

total number of trees per acre = = 200

the yield will decrease by : -

⇒total yield =

B.

⇒to maximize the yield ,

let's take the number of trees per acre to be 100+y ;

and thus the average yield per acre = 150 - y;

total yield =

this is a quadratic equation. this can be rewritten as ,

     ⇒  

In this equation , the total yield becomes maximum when y=25;

⇒Thus the total number of trees per acre = 100+25 =125;

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4