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.
<h3>How to determine the true profit functions</h3>
From the complete questions, the profit functions are calculated as follows:
<u>Material 1</u>
![P(x) = 5000000- 20000x - 200000x + 2000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%205000000-%2020000x%20-%20200000x%20%2B%202000x%5E2)
![P(x) = 5000000-220000x + 2000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%205000000-220000x%20%2B%202000x%5E2)
<u>Material 2</u>
![P(x) = 4000000 - 10000x - 160000x+ 1000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%204000000%20-%2010000x%20-%20160000x%2B%201000x%5E2)
![P(x) = 4000000-170000x+ 1000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%204000000-170000x%2B%201000x%5E2)
<u>Material 3</u>
![P(x) = 2000000 - 5000x - 54000x - 270x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%202000000%20-%205000x%20-%2054000x%20-%20270x%5E2)
![P(x) = 2000000 -59000x - 270x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%202000000%20-59000x%20-%20270x%5E2)
Next, we test the options
Option 1: When material 2 is used
A price is greater than $150 is $151.
Calculate P(151) using ![P(x) = 4000000-170000x+ 1000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%204000000-170000x%2B%201000x%5E2)
So, we have:
![P(151) = 4000000-170000 * 151+ 1000* 151^2](https://tex.z-dn.net/?f=P%28151%29%20%3D%204000000-170000%20%2A%20151%2B%201000%2A%20151%5E2)
![P(151) = 1131000](https://tex.z-dn.net/?f=P%28151%29%20%3D%201131000)
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 ![P(x) = 5000000-220000x + 2000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%205000000-220000x%20%2B%202000x%5E2)
So, we have:
![P(49) = 5000000-220000 * 49 + 2000* 49^2](https://tex.z-dn.net/?f=P%2849%29%20%3D%205000000-220000%20%2A%2049%20%2B%202000%2A%2049%5E2)
![P(49) =-978000](https://tex.z-dn.net/?f=P%2849%29%20%3D-978000)
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 ![P(x) = 2000000 -59000x - 270x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%202000000%20-59000x%20-%20270x%5E2)
So, we have:
![P(45) = 2000000 -59000 * 45 - 270 * 45^2](https://tex.z-dn.net/?f=P%2845%29%20%3D%202000000%20-59000%20%2A%2045%20-%20270%20%2A%2045%5E2)
![P(45) = -1201750](https://tex.z-dn.net/?f=P%2845%29%20%3D%20-1201750)
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 ![P(x) = 4000000-170000x+ 1000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%204000000-170000x%2B%201000x%5E2)
So, we have:
![P(30) = 4000000-170000*30+ 1000*30^2](https://tex.z-dn.net/?f=P%2830%29%20%3D%204000000-170000%2A30%2B%201000%2A30%5E2)
![P(30) = -200000](https://tex.z-dn.net/?f=P%2830%29%20%3D%20-200000)
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 ![P(x) = 5000000-220000x + 2000x^2](https://tex.z-dn.net/?f=P%28x%29%20%3D%205000000-220000x%20%2B%202000x%5E2)
So, we have:
![P(40) = 5000000-220000*40 + 2000*40^2](https://tex.z-dn.net/?f=P%2840%29%20%3D%205000000-220000%2A40%20%2B%202000%2A40%5E2)
![P(40) = -600000](https://tex.z-dn.net/?f=P%2840%29%20%3D%20-600000)
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:
brainly.com/question/25638609