Question

In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x items. They estimate the cost to produce 1,000 items/day as $54,745, and the cost to produce 5,000 items/day as $73,731.

a)Find a formula for c(x).
b)find the fixed cost.
c)find the total cost to produce 100,000 items.
d)find the marginal cost of the items to be produced in this plant and what does this mean to the manager?

Answer 1

Answer:

a) $C(x)=4.7465x+49988.5$

b) $49988.5

c) $524638.50

d) Marginal cost: $4.75.

Step-by-step explanation:

We have been given that a manufacturing company estimate the cost to produce 1,000 items/day as $54,745, and the cost to produce 5,000 items/day as $73,731.

a) First of all, we will find slope of line using points (1000,54745) and (5000,73731).

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{73731-54745}{5000-1000}$

$m=\frac{18986}{4000}$

$m=4.7465$

Now, we will write our equation in point slope form using point (1000,54745) and $m=4.7465$ as:

$y-y_1=m(x-x_1)$

$y-54745=4.7465(x-1000)$

$y-54745=4.7465x-4756.5$

$y=4.7465x-4756.5+54745$

$y=4.7465x+49988.5$

Therefore, our required cost function would be $C(x)=4.7465x+49988.5$.

b) We can see that our cost function is in slope-intercept form, where 49988.50 represents initial amount or fixed cost, therefore, fixed cost would be $49988.50.

c) To find the total cost to produce 100,000 items, we will substitute $x=100,000$ in our cost function as:

$C(100,000)=4.7465(100,000)+49988.5$

$C(100,000)=474,650+49,988.5\\\\C(100,000)=524,638.5$

Therefore, the total cost to produce 100,000 items would be $524638.50.

d) We know that marginal cost is the cost added by producing one additional unit of a product or service.

We know that variable cost of each product is $4.7465 or approximately $4.75. Therefore, marginal cost of the items to be produced is $4.75 and it means that for every one unit increase in production, the manager has to spend $4.75.