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Sagot :
The profit function is the difference between the cost and the revenue functions.
The true statement is (a) If material 2 is used, Carrie will earn a profit if she sells chairs for more than $150 each.
How to determine the true profit functions
From the complete questions, the profit functions are calculated as follows:
Material 1
[tex]P(x) = 5000000- 20000x - 200000x + 2000x^2[/tex]
[tex]P(x) = 5000000-220000x + 2000x^2[/tex]
Material 2
[tex]P(x) = 4000000 - 10000x - 160000x+ 1000x^2[/tex]
[tex]P(x) = 4000000-170000x+ 1000x^2[/tex]
Material 3
[tex]P(x) = 2000000 - 5000x - 54000x - 270x^2[/tex]
[tex]P(x) = 2000000 -59000x - 270x^2[/tex]
Next, we test the options
Option 1: When material 2 is used
A price is greater than $150 is $151.
Calculate P(151) using [tex]P(x) = 4000000-170000x+ 1000x^2[/tex]
So, we have:
[tex]P(151) = 4000000-170000 * 151+ 1000* 151^2[/tex]
[tex]P(151) = 1131000[/tex]
P(151) is greater than 0; this represents a profit
Hence, option (1) is true
Option 2: When material 1 is used
A price is less than $50 is $49.
Calculate P(49) using [tex]P(x) = 5000000-220000x + 2000x^2[/tex]
So, we have:
[tex]P(49) = 5000000-220000 * 49 + 2000* 49^2[/tex]
[tex]P(49) =-978000[/tex]
P(49) is less than 0; this represents loss
Hence, option (2) is false
Option 3: When material 3 is used
Calculate P(45) and P(160) using [tex]P(x) = 2000000 -59000x - 270x^2[/tex]
So, we have:
[tex]P(45) = 2000000 -59000 * 45 - 270 * 45^2[/tex]
[tex]P(45) = -1201750[/tex]
P(45) is less than 0; this represents loss
Hence, option (3) is false
Option 1: When material 2 is used
Calculate P(30) and P(120) using [tex]P(x) = 4000000-170000x+ 1000x^2[/tex]
So, we have:
[tex]P(30) = 4000000-170000*30+ 1000*30^2[/tex]
[tex]P(30) = -200000[/tex]
P(30) is less than 0; this represents a loss
Hence, option (4) is false
Option 2: When material 1 is used
Calculate P(40) and P(70) using [tex]P(x) = 5000000-220000x + 2000x^2[/tex]
So, we have:
[tex]P(40) = 5000000-220000*40 + 2000*40^2[/tex]
[tex]P(40) = -600000[/tex]
P(49) is less than 0; this represents loss
Hence, option (5) is false
The above means that:
The true statement is (a) If material 2 is used, Carrie will earn a profit if she sells chairs for more than $150 each.
Read more about revenue functions at:
https://brainly.com/question/25638609
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