example 3 Peanut Production

Chapter 4.2

Georgia’s production of peanuts has increased moderately in the last 10 years, but profit margins have been reduced by lower prices, declining yields, and increasing costs. The table gives the annual revenue, variable costs, and fixed costs for peanut production in Georgia for the years 1998–2002.

1998 1999 2000 2001 2002

Revenue ($) 57,250 58,395 59,563 60,754 61,969

Variable Costs ($) 28,064 28,905 29,773 30,666 31,586

Fixed Costs ($) 11,382 11,724 12,076 12,438 12,811

The function that models the revenue from peanut production in Georgia for theyears 1998–2002 is given by

where x is number of years after 1990. The function that models the total cost (variableplus fixed) of peanut production in Georgia is given by

where x is the number of years after 1990.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

2009 PBLPathways

2009 PBLPathways

Georgia’s production of peanuts has increased moderately in the last 10 years, but profit margins have been reduced by lower prices, declining yields, and increasing costs. The table gives the annual revenue, variable costs, and fixed costs for peanut production in Georgia for the years 1998–2002.

1998 1999 2000 2001 2002

Revenue ($) 57,250 58,395 59,563 60,754 61,969

Variable Costs ($) 28,064 28,905 29,773 30,666 31,586

Fixed Costs ($) 11,382 11,724 12,076 12,438 12,811

The function that models the revenue from peanut production in Georgia for theyears 1998–2002 is given by

where x is number of years after 1990. The function that models the total cost (variableplus fixed) of peanut production in Georgia is given by

where x is the number of years after 1990.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

b. Graph the functions R, C, and P on the same set of axes.

c. What is the slope of the graph of P(x)? Interpret the slope as a rate of change.

d. If the model remains accurate, what will be the profit for peanut production in 2010?

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

1.1797 47.789( ) thousand d l2 o larsxR x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

1.1797 47.789( ) thousand d l2 o larsxR x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.2377 29.508

1.1797 47.789

1.1797 47.78

2 1.237

9

7 29.508

0.058 18 281

2

. 2

P x R x C x

x

x x

x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand doll arsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand doll arsR x x

1.2377 29.508( ) thousand dollarsC xx

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892

1.1797 47.7892 1

1.

.2

2377

377 29.508

0.058 1

29.508

8.2812

P x R x C x

x

x x

x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand doll arsR x x

1.2377 29.508( ) thousand dollarsC xx

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892

1.1797 47.7892 1

1.

.2

2377

377 29.508

0.058 1

29.508

8.2812

P x R x C x

x

x x

x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x ?

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

a. Write a function P(x) that models the profit from peanut production.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

( ) ( ) ( )

1.1797 47.7892 1.2377 29.508

1.1797 47.7892 1.2377 29.508

0.058 18.2812

P x R x C x

x x

x x

x

2009 PBLPathways

b. Graph the functions R, C, and P on the same set of axes.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

2009 PBLPathways

b. Graph the functions R, C, and P on the same set of axes.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

R(x)

C(x)

P(x)

2009 PBLPathways

c. What is the slope of the graph of P(x)? Interpret the slope as a rate of change.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

( )P x mx b

m = -0.058 b = 18.2812

-0.058 thousand dollars per year or -58 dollars per year

2009 PBLPathways

c. What is the slope of the graph of P(x)? Interpret the slope as a rate of change.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

( )P x mx b

m = -0.058 b = 18.2812

-0.058 thousand dollars per year or -58 dollars per year

2009 PBLPathways

c. What is the slope of the graph of P(x)? Interpret the slope as a rate of change.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

( )P x mx b

m = -0.058 b = 18.2812

-0.058 thousand dollars per year or -58 dollars per year

2009 PBLPathways

c. What is the slope of the graph of P(x)? Interpret the slope as a rate of change.

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

( )P x mx b

m = -0.058 b = 18.2812

-0.058 thousand dollars per year or -58 dollars per year

2009 PBLPathways

d. If the model remains accurate, what will be the profit for peanut production in 2010?

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

(20) 0.058 20 18.2812

17.1212 thousand dollars

P

2009 PBLPathways

d. If the model remains accurate, what will be the profit for peanut production in 2010?

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

(20) 0.058 20 18.2812

17.1212 thousand dollars

P

2009 PBLPathways

d. If the model remains accurate, what will be the profit for peanut production in 2010?

( ) 1.1797 47.7892 thousand dollarsR x x

( ) 1.2377 29.508 thousand dollarsC x x

( ) 0.058 18.2812 thousand dollarsP x x

(20) 0.058 20 18.2812

17.1212 thousand dollars

P