Lorimar Watch Company manufactures travel clocks. The daily...

Lorimar Watch Company manufactures travel clocks. The daily marginal cost function associated with producing these clocks is $$ C^{\prime}(x)=0.000009 x^{2}-0.009 x+8 $$ where $C^{\prime}(x)$ is measured in dollars/unit and $x$ denotes the number of units produced. Management has determined that the daily fixed cost incurred in producing these clocks is $\$ 120 .$ Find the total cost incurred by Lorimar in producing the first 500 travel clocks/day.

Lorimar Watch Company manufactures travel clocks. The daily marginal cost function associated with producing these clocks is
$$
C^{\prime}(x)=0.000009 x^{2}-0.009 x+8
$$
where $C^{\prime}(x)$ is measured in dollars/unit and $x$ denotes the number of units produced. Management has determined that the daily fixed cost incurred in producing these clocks is $\$ 120 .$ Find the total cost incurred by Lorimar in producing the first 500 travel clocks/day.

Key Concepts

Marginal Cost Function

The marginal cost function represents the rate at which the total production cost changes with respect to a change in production quantity. It is mathematically the derivative of the total cost function with respect to the number of units produced. This concept is central in production and economic analysis as it helps determine how costs evolve as production scales up.

Definite Integration

Definite integration is a fundamental calculus technique used to compute the accumulation of quantities, such as costs. In the context of cost analysis, integrating the marginal cost function over a specified interval gives the total variable cost incurred over that production range. This method transforms the rate of cost change back into the total cost form.

Fixed Costs

Fixed costs are expenses that do not vary with the level of production output and must be incurred regardless of how many items are produced. They represent the baseline expenses associated with running a production process, such as rent, salaries, and other overheads, and are additive to the variable costs calculated from the marginal cost function.

Total Cost Calculation

The total cost function combines both the variable cost, obtained by integrating the marginal cost function over the production interval, and the fixed costs. By summing these two components, one can determine the overall cost incurred for a given level of production, which is essential for budgeting and economic decision-making.

Transcript

00:01 A company that manufactures travel clocks has determined that the daily marginal cost function associated with producing these clocks is derivative of c of x equals 0 .009 times x squared minus 0 .009 times x plus 8, where c derivative of x is measured in dollars, per unit and x denotes the number of units produced.

00:37 Management has also determined that the daily fixed cost incurring producing these clocks is $120.

00:46 And with that information, we want to find the total cost incurred by the company in producing the first 500 travel clocks per day.

01:00 So we have the marginal cost function that is the derivative of the cost function and we want to find the total code so we can't find the antiderivative or indefinite integral of the c prime so we know that total cost c of x will be equal to the integral of the marginal coast and that will be equal to the integral the indefinite integral of the expression given to the marginal cost that is 0 .009 x squared then minus 0 .009 x plus 8 differential of x.

02:14 And we got to integrate this and this is a constant so we get 0 .0000009 times the of x square minus 0 .009 which is a constant times the integral of x plus 8 integral of differential of x and these are powers of x and we know how to integrate that is 0 .009 x to the 3x minus 0 .009 x to the 3 minus 0 .000 .009 times x to the 3 square over 2 plus 8x plus a constant of integration.

03:11 We put the constant of integration at the end because even if each integral has a constant of integration, we know that the sum of them will be a unique constant of integration.

03:23 So the character of being indefinite is the reason why we have a constant of integration, which has to be determined with some initial condition.

03:34 Let's simplify a little bit here.

03:37 So we have 0 .00009 over 3 is 0 .000000x cubed minus 0 .009 over 2 is 0 .0045 x squared plus 8x plus c.

04:01 And this is the total cost function.

04:04 With this we can calculate the total cost for any number of units, but before we get to determine this value, the value of this constant of integration.

04:17 And for that, we have the initial value given here...

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