A company manufactures widgets. The daily marginal cost to produce x widgets is found to be

C'(x) = 0.000009x^2 - 0.009x + 8

(measured in dollars per unit). The daily fixed costs are found to be $120.

a. Use this information to get a general cost function for producing widgets.

b. Find the total cost of producing the first 500 widgets.

c. If you sell the widgets for $25 each, how many will need to be sold before the company begins making a profit? (Hint: The revenue function is R(x) = $25x;
after you integrate C’(x), set C(x) = R(x) and solve for x. Finding the
intersection of the graphs of these two function may help you here.)

c = 9*10^-6 x^3/3 - 9*10^-3 x^2/2 + 8 x + constant which is 120 so

c = 3 *10^-6 x^3 -4.5*10^-3 x^2 + 8 x + 120
if x = 5*10^2
go ahead, plug that in

then for break even
25 x = = 3 *10^-6 x^3 -4.5*10^-3 x^2 + 8 x + 120

or
0 = 3*10^-6 x^3 -4.5*10^-3 x - 17 x +120

Using an online roots calculator I get x = 3243.358, -1750.404, 7.046

so perhaps 7
check my arithmetic!

a. To find the general cost function for producing widgets, we need to integrate the marginal cost function C'(x).

C(x) = ∫ C'(x) dx

C(x) = ∫ (0.000009x^2 - 0.009x + 8) dx

C(x) = 0.000003x^3 - 0.0045x^2 + 8x + C

We can determine the constant of integration, C, by using the information that the daily fixed costs are $120. So, when x = 0, C(x) should be equal to $120.

C(0) = 0.000003(0)^3 - 0.0045(0)^2 + 8(0) + C

C(0) = C

C = 120

Therefore, the general cost function for producing widgets is:

C(x) = 0.000003x^3 - 0.0045x^2 + 8x + 120

b. To find the total cost of producing the first 500 widgets, we substitute x = 500 into the cost function C(x).

C(500) = 0.000003(500)^3 - 0.0045(500)^2 + 8(500) + 120

C(500) = 0.000003(125000000) - 0.0045(250000) + 4000 + 120

C(500) = 375 - 1125 + 4120 + 120

C(500) = $3490

Therefore, the total cost of producing the first 500 widgets is $3490.

c. To find the number of widgets that need to be sold before the company begins making a profit, we set the cost function C(x) equal to the revenue function R(x) and solve for x.

C(x) = R(x)
0.000003x^3 - 0.0045x^2 + 8x + 120 = 25x

Rearranging the equation:
0.000003x^3 - 0.0045x^2 + 8x + 120 - 25x = 0

Combining like terms:
0.000003x^3 - 0.0045x^2 - 17x + 120 = 0

This equation is a cubic equation and may require numerical methods to solve. However, one way to solve this equation is by finding the intersection of the graphs of these two functions using a graphing calculator or software. The value of x at the intersection point represents the number of widgets that need to be sold before the company begins making a profit.

a. To obtain the general cost function, we need to integrate the marginal cost function, C'(x), with respect to x. Since the marginal cost function is given as:

C'(x) = 0.000009x^2 - 0.009x + 8

we can integrate term by term to find the cost function, C(x). The integral of x^2 with respect to x is (1/3)x^3, the integral of -0.009x with respect to x is -0.009(1/2)x^2 = -0.0045x^2, and the integral of 8 with respect to x is 8x. Adding these results together, we find:

C(x) = (1/3)x^3 - 0.0045x^2 + 8x

This is the general cost function for producing widgets.

b. To find the total cost of producing the first 500 widgets, we substitute x = 500 into the cost function, C(x). Therefore, we have:

C(500) = (1/3)(500)^3 - 0.0045(500)^2 + 8(500)
= (1/3)(125,000,000) - 0.0045(250,000) + 4,000
= 41,666,666.67 - 1,125 + 4,000
= 41661941.67

The total cost of producing the first 500 widgets is approximately $41,661,941.67.

c. To find the number of widgets that need to be sold before the company begins making a profit, we need to set the cost function, C(x), equal to the revenue function, R(x), and solve for x. The revenue function is given as:

R(x) = $25x

Therefore, we have the equation:

C(x) = R(x)
(1/3)x^3 - 0.0045x^2 + 8x = 25x

To solve this equation, we subtract 25x from both sides:

(1/3)x^3 - 0.0045x^2 + 8x - 25x = 0

Combine like terms:

(1/3)x^3 - 0.0045x^2 - 17x = 0

Now, we need to find the value of x that satisfies this equation. This can be done using numerical methods, such as graphing the two functions and finding their intersection point, or using a calculator or software that can solve equations. The solution to this equation will give us the number of widgets that need to be sold before the company begins making a profit.