arithmetic and composition (please help ive tried everything!)

Question

The cost to manufacture a product is proportional to the quantity produced, with a cost of $5000 dollars per day when 100 items are produced. The cost to store n items is proportional to the square of n, with a daily storage cost of $60 dollars when 100 items are stored. In addition, there is a fixed daily cost of $10000. The total daily cost, T when producing and storing n items is the sum of production, storage and fixed costs.

Write a formula for T as a function of n.

Answer 1

Let's look at this one piece at a time. First, we are told, "The cost to manufacture a product is proportional to the quantity produced." This can be expressed mathematically as:

P ∝ n

where M is the cost to manufacture and n is the number produced. We can turn this into an equation by introducing an unknown constant of proportionality, which well call a:

P = a n.

We can find the value of a by using the next piece of information that is given, ". . . a cost of $5000 dollars per day when 100 items are produced"

5000 = a ×100,

a = 50

P = 50 n

We can repeat the same process with the next two items, "The cost to store n items is proportional to the square of n":

S ∝ n²

S = b n²

"a daily storage cost of $60 dollars when 100 items are stored"

60 = b ×100²

b = 0.006

S = 0.006 n²

"a fixed daily cost of $10000"

F = 10000

The total daily cost, T when producing and storing n items is the sum of production [P], storage [S] and fixed costs [F].

T = P + S + F

T = 50 n + 0.006 n² +10000

Answer 2

production cost = 5000/100 = $50 per item

cost to store n items = k*n^2

60 = k*100^2

60/10000 = k

6/1000 =k

3/500 = k

so storage cost = (3/500)*n^2

while the product cost is : 50n

T(n) = 50n + (3/600)n^2 + 10000