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Intuitive Calculus: Maximizing Orange Harvest, Exams of Calculus

In this document from math 11012: intuitive calculus, spring 2011, students are asked to find the optimal number of orange trees to plant per acre to maximize the total yield. The problem involves creating expressions for the number of trees per acre and yield per tree based on the given information, then using calculus to find the maximum yield. Suitable for university students taking calculus courses.

What you will learn

  • What expression represents the yield per tree based on the given information?
  • How do you find the absolute maximum yield using calculus?
  • What expression represents the number of trees per acre based on the given information?

Typology: Exams

2019/2020

Uploaded on 06/18/2020

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MATH 11012 Intuitive Calculus Spring 2011
Circle one: 8:50 5:30 Ms. Kracht
Name: Quiz Score: /25
Quiz 3: Version A
Show your reasoning. Use standard notation correctly. Simplify your answers.
You may NOT share calculators or use a cell phone or any Internet device as a calculator.
1. An orange grower finds that if she plants 30 trees per acre, each tree will yield 70 bushels of oranges. She also
estimates that for each additional tree that she plants per acre, the yield of each tree will decrease by 5 bushels. We
wish to find how many trees she should plant per acre to maximize her harvest (total yield per acre).
Let xbe the number of additional trees she plants per acre.
(a) (4 pts) Find an expression in terms of xfor the number of trees per acre.
(b) (4 pts) Find an expression in terms of xfor the yield per tree.
(c) (4 pts) Write a function Y(x)representing the total yield per acre. Simplify.
(d) (6 pts) Use calculus to find the value of the variable which gives the absolute maximum value of your yield
function.
(e) (4 pts) Verify that this is the indeed the absolute maximum on the appropriate interval. Explain!
(f) (3 pts) Answer the question in a complete sentence.
pf2

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MATH 11012 Intuitive Calculus Spring 2011 Circle one: 8:50 5:30 Ms. Kracht

Name: Quiz Score: /

Quiz 3: Version A

Show your reasoning. Use standard notation correctly. Simplify your answers. You may NOT share calculators or use a cell phone or any Internet device as a calculator.

  1. An orange grower finds that if she plants 30 trees per acre, each tree will yield 70 bushels of oranges. She also estimates that for each additional tree that she plants per acre, the yield of each tree will decrease by 5 bushels. We wish to find how many trees she should plant per acre to maximize her harvest (total yield per acre).

Let x be the number of additional trees she plants per acre.

(a) (4 pts) Find an expression in terms of x for the number of trees per acre.

(b) (4 pts) Find an expression in terms of x for the yield per tree.

(c) (4 pts) Write a function Y (x) representing the total yield per acre. Simplify.

(d) (6 pts) Use calculus to find the value of the variable which gives the absolute maximum value of your yield function.

(e) (4 pts) Verify that this is the indeed the absolute maximum on the appropriate interval. Explain!

(f) (3 pts) Answer the question in a complete sentence.