Let R(x), C(x), and P(x) be, respectively, the revenue, cost, and profit, in dollars, from the production and sale of x it
ems. If R(x) = 6x and C(x) equals = 0.002x^2+2.2x+40, find each of the following. a. p(x) b.p(100) c. P'(x) d.P'(100)
1 answer:
Answer:
![a.\ P(x) = - 0.002x^2+3.8x-40](https://tex.z-dn.net/?f=a.%5C%20P%28x%29%20%3D%20-%200.002x%5E2%2B3.8x-40)
b. 320
![c.\ P'(x) = -0.004 x + 3.8](https://tex.z-dn.net/?f=c.%5C%20P%27%28x%29%20%3D%20-0.004%20x%20%2B%203.8)
d. 3.76
Step-by-step explanation:
Given that:
Revenue function:
![R(x) = 6x](https://tex.z-dn.net/?f=R%28x%29%20%3D%206x)
Cost Function:
![C(x) = 0.002x^2+2.2x+40](https://tex.z-dn.net/?f=C%28x%29%20%3D%200.002x%5E2%2B2.2x%2B40)
We know that,
Answer a: Profit = Revenue - Cost
![\Rightarrow P(x) = R(x) - C(x)\\\Rightarrow P(x) = 6x - (0.002x^2+2.2x+40)\\\Rightarrow P(x) = - 0.002x^2+3.8x-40](https://tex.z-dn.net/?f=%5CRightarrow%20P%28x%29%20%3D%20R%28x%29%20-%20C%28x%29%5C%5C%5CRightarrow%20P%28x%29%20%3D%206x%20-%20%280.002x%5E2%2B2.2x%2B40%29%5C%5C%5CRightarrow%20P%28x%29%20%3D%20-%200.002x%5E2%2B3.8x-40)
Answer b:
P(100) = ?
Putting value of x as 100 in above equation:
![P(100) = - 0.002\times 100^2+3.8 \times 100-40\\\Rightarrow P(100) = -20+380 -40\\\Rightarrow P(100) = 320](https://tex.z-dn.net/?f=P%28100%29%20%3D%20-%200.002%5Ctimes%20100%5E2%2B3.8%20%5Ctimes%20100-40%5C%5C%5CRightarrow%20P%28100%29%20%3D%20-20%2B380%20-40%5C%5C%5CRightarrow%20P%28100%29%20%3D%20320)
Answer c:
P'(x) = ?
Differentiating the equation ![P(x) = - 0.002x^2+3.8x-40](https://tex.z-dn.net/?f=P%28x%29%20%3D%20-%200.002x%5E2%2B3.8x-40)
![P'(x) = -2 \times 0.002 x^{2-1} + 3.8 + 0\\P'(x) = -0.004 x + 3.8](https://tex.z-dn.net/?f=P%27%28x%29%20%3D%20-2%20%5Ctimes%200.002%20x%5E%7B2-1%7D%20%2B%203.8%20%2B%200%5C%5CP%27%28x%29%20%3D%20-0.004%20x%20%2B%203.8)
Answer d:
P'(100) = ?
Putting x = 100 in equation ![P'(x) = -0.004 x + 3.8](https://tex.z-dn.net/?f=P%27%28x%29%20%3D%20-0.004%20x%20%2B%203.8)
![P'(100) = -0.004 \times 100 + 3.8\\P'(100) = -0.4 + 3.8\\P'(100) = 3.76](https://tex.z-dn.net/?f=P%27%28100%29%20%3D%20-0.004%20%5Ctimes%20100%20%2B%203.8%5C%5CP%27%28100%29%20%3D%20-0.4%20%2B%203.8%5C%5CP%27%28100%29%20%3D%203.76)
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