Question 1149820: A farmer has 1,080 acres of land on which he grows corn, wheat, and soybeans. It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans. Because of market demand the farmer will grow twice as many acres of wheat as of corn. He has allocated $57,300 for the cost of growing his crops. How many acres of each crop should he plant?
Found 2 solutions by ankor@dixie-net.com, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A farmer has 1,080 acres of land on which he grows corn, wheat, and soybeans.
It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans.
Because of market demand the farmer will grow twice as many acres of wheat as of corn.
He has allocated $57,300 for the cost of growing his crops.
How many acres of each crop should he plant?
:
let c = acres in corn
let w = acres in wheat
let s = acres in soy bean
:
write an equation for each statement
:
"A farmer has 1,080 acres of land on which he grows corn, wheat, and soybeans."
c + w + s = 1080
" It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans.".."He has allocated $57,300 for the cost"
45c + 60w + 50s = 57300
"Because of market demand the farmer will grow twice as many acres of wheat as of corn."
w = 2c
:
replace w with 2c in the first two equations
c + 2c + s = 1080
3c + s = 1080
and
45c + 60(2c) + 50s = 57300
45c + 120c + 50s = 57300
165c + 50s = 57300
simplify, divide equation by 50
3.3c + s = 1146
use elimination on these two equation
3.3c + s = 1146
3c + s = 1080
--------------------subtraction eliminates s, find c
.3c + 0 = 66
c = 66/.3
c = 220 acres in corn
then
w = 2(220)
w = 440 acres in wheat
then
1080 - 220 - 440 = 420 acres in soybeans
:
:
Check this in the statement:
" It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans.".."He has allocated $57,300 for the cost"
45(220) + 60(440) + 50(420) =
9900 + 26400 + 21000 = 57300
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! .
There is much simpler way to solve the problem, using one SINGLE equation in one unknown
INSTEAD of system of 3 equations in 3 unknowns.
Let x be the area for corn, in acres.
Then the area for wheat is 2x acres.
Hence, the area for soybeans is (1080 - x - 2x) = 1080-3x acres.
Now it is easy to write the "money" equation
45*x + 60*(2x) + 50*(1080-3x) = 57300.
Simplify and solve for x
45x + 120x + 50*1080 - 150x = 57300
15x = 57300 - 50*1080 = 3300
x = 3300/15 = 220.
ANSWER. 220 acres for corn; 2*220 = 440 acres for wheat and the rest, 1080-220-440 = 420 acres for soybeans.
Solved and completed.
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The whole problem is designed, intended and expected to be solved exactly IN THIS WAY.
It was given to you WITH THE ONLY GOAL : in order for you learned from my post how to solve it in the RIGHT WAY.
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