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.

To find the profit function, P(x), we need to subtract the total cost function, C(x), from the total revenue function, R(x):

P(x) = R(x) - C(x)

Given that:

R(x) = 47x - 0.3x^2

C(x) = 8x + 16

We can substitute these values into the equation for P(x):

P(x) = (47x - 0.3x^2) - (8x + 16)

Now, let's simplify this equation:

P(x) = 47x - 0.3x^2 - 8x - 16

Combining like terms, we get:

P(x) = -0.3x^2 + 39x - 16

To find the profit for 100 items produced, we substitute x = 100 into the profit function:

P(100) = -0.3(100)^2 + 39(100) - 16

P(100) = -0.3(10000) + 3900 - 16

P(100) = -3000 + 3900 - 16

P(100) = 884

So, the profit for producing 100 items is $884.

To find the number of items that maximize the profit, we need to determine the value of x that gives the maximum value for the profit function, P(x).

Since the profit function is a quadratic equation in the form ax^2 + bx + c, we know that the maximum (or minimum) value occurs at the vertex of the parabolic curve.

The x-coordinate of the vertex is given by the formula: x = -b / (2a).

For our profit function, the values of a, b, and c are:

a = -0.3, b = 39, c = -16

Substituting these values into the vertex formula, we get:

x = -39 / (2 * -0.3)

x = -39 / -0.6

x = 65

So, to maximize the profit, you need to produce 65 items.

Please note that in this context, we assume that production levels can be fractional or continuous. If production can only be done in whole numbers, the closest whole number to 65 (either 64 or 66) would be considered.