At Allied electronics, production has begun on x-15 computer chip. The total revenue function is given by. R(x)=47x-0.3x^2 and the total cost function is given C(x)=8x+16, where x represents the number of boxes of computer chip produced.
A. Find the profit function in P(x)=R(x)-C(x)
Find the profit for 100 items produced
How many items do you have to produce to maximize the profit.
Thanks for your help.
A. They've told you what P(x) is:
P(x) = R(x)-C(x)
= 47x-.3x^2 - (8x+16)
= -.3x^2 + 39x - 16
Now , that's just a parabola, and you know that the parabola ax^2+bx+c has its vertex at x = -b/2a, so
max profit P(x) occurs at
x = -39/-.6 = 65
A. To find the profit function, subtract the total cost function C(x) from the total revenue function R(x):
P(x) = R(x) - C(x)
P(x) = (47x - 0.3x^2) - (8x + 16)
P(x) = 47x - 0.3x^2 - 8x - 16
Simplifying,
P(x) = -0.3x^2 + 39x - 16
To find the profit for 100 items produced, substitute x = 100 into the profit function P(x):
P(100) = -0.3(100)^2 + 39(100) - 16
P(100) = -0.3(10,000) + 3,900 - 16
P(100) = -3,000 + 3,900 - 16
P(100) = 884
The profit for producing 100 items is $884.
To find the number of items (x) that maximizes profit, we need to find the vertex of the profit function P(x). The x-coordinate of the vertex gives us the number of items to produce to maximize profit.
The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In our case, a = -0.3 and b = 39.
x = -39 / (2(-0.3))
x = -39 / (-0.6)
x = 65
Therefore, to maximize profit, you need to produce 65 items.
A. To find the profit function, P(x), subtract the total cost function, C(x), from the total revenue function, R(x):
P(x) = R(x) - C(x)
Substituting the given functions:
P(x) = (47x - 0.3x^2) - (8x + 16)
P(x) = 47x - 0.3x^2 - 8x - 16
Simplifying further:
P(x) = -0.3x^2 + 39x - 16
Now that we have the profit function, we can move on to the other questions.
To find the profit for 100 items produced, we can substitute x = 100 into the profit function P(x):
P(100) = -0.3(100)^2 + 39(100) - 16
P(100) = -0.3(10000) + 3900 - 16
P(100) = -3000 + 3900 - 16
P(100) = 884
Therefore, the profit for producing 100 items is $884.
To determine the number of items that will maximize the profit, we need to find the vertex of the quadratic profit function. The x-coordinate of the vertex gives us the number of items to produce.
Given the profit function: P(x) = -0.3x^2 + 39x - 16
The vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b / (2a). In this case, a = -0.3 and b = 39.
x = -39 / (2 * -0.3)
x = -39 / (-0.6)
x = 65
Therefore, to maximize the profit, you need to produce 65 items.
I hope this explanation helps! Let me know if there's anything else I can assist you with.