8. An orchard..

presently has 30 trees per acre. Each tree yields 400 peaches. It is determined that each additional

tree that is planted per acre, the yield will be reduced by 20 peaches per tree. How many trees should be added

per acre to maximize the yield?

What is the unknown?

What are the given quantities?

What are the given conditions?

Notations

P(t)= (400-20t)(30+t)

We want to optimize the production of peaches in relation to the number of trees in an acre.

Peach Production

P-Peach production to be maximized when t trees are added.

t-trees

We know, the amount of peaches depends on the amount of trees,therefore, :

t- independent variable

p- dependent variable

THEREFORE:

P(t)=(400-20t)(30+t)

T-CHART

Proving the optimization derivative...

-40t-200=0

-40t=200

t=-5

Derivatives

For our peach production optimization

P'(t)=-40t-200

**Math Partial 1 Project**

UNKNOWN

Amount of trees we needed to maximize the amount of peaches produced per acre.

GIVEN DATA

Amount of trees per acre:

30 trees

Amount of peaches per tree:

400 peaches

Amount of peaches per tree when there are additional trees (per acre):

+1 tree-20peaches

Conditions

When a tree is added to the acre (having 30 trees), the production of peaches per tree would decrease in 20 peaches.

P(t)= -20t -200t+12000

2

**Diana Cristina Castro Armenta A01225990**

Luz América Paczka Giorgi A01226178

Group 11

Mathematics AP

Professor: Tiberiu Hrabak

Luz América Paczka Giorgi A01226178

Group 11

Mathematics AP

Professor: Tiberiu Hrabak

P(t)= -20t -200t+12000

2

P''(t)=-40

P'(t)=-40t-200

P(t)

t

-5

*

P(t)= -20t -200t+12000

2

-8

0

12320

12500

12000

GRAPH

t

p

P(-5)= [400-20(-5)](30-5)

P(-5)=500*25

P= 12500

To prove...

After reading and understanding the problem...

P'(t)

t

-5

-8

0

120

0

-200