Question

For the given cost function C ( x ) = 19600 + 500 x + x 2 , First, find the average cost function. Use it to find: a) The production level that will minimize the average cost

Answer 1

To find the average cost function, divide the total cost by the production level. To find the production level that will minimize the average cost, take the derivative of the average cost function, set it equal to zero, and solve for x. The production level that will minimize the average cost is √19600.

Step-by-step explanation:

To find the average cost function, we need to divide the total cost by the production level. The average cost function is given by:

AC(x) = C(x)/x

where C(x) is the cost function.

To find the production level that will minimize the average cost, we need to take the derivative of the average cost function, set it equal to zero, and solve for x.

Let's go through the steps:

1. Divide the cost function by the production level: AC(x) = (19600 + 500x + x^2) / x
2. Simplify the expression: AC(x) = 19600/x + 500 + x
3. Take the derivative of AC(x): AC'(x) = -19600/x^2 + 1
4. Set AC'(x) equal to zero and solve for x: -19600/x^2 + 1 = 0
5. Multiply both sides by x^2: -19600 + x^2 = 0
6. Solve the quadratic equation for x: x^2 = 19600
7. Take the square root of both sides: x = ±√19600
8. Since x represents the production level, it cannot be negative. So, x = √19600

Therefore, the production level that will minimize the average cost is √19600.

Answer 2

The answer is "140".

Step-by-step explanation:

Calculating the average cost:

`= (C(x))/(x)\\\\ = (19600+500x+x^2)/(x) \\\\ = (19600)/(x)+500 +x\\\\`

To minimize average cost we have to find x for wich A'(x)=0

`A(x) = (19600)/(x)+500 +x \\\\A'(x) = -(19600)/(x^2)+1 \\\\0= -(19600)/(x^2)+1 \\\\(19600)/(x^2)=1 \\\\19600=x^2 \\\\x=140`