Question

The cost to manufacture a product is proportional to the quantity produced, with a cost of $35000 dollars per day when 700 items are produced. The cost to store n items is proportional to the square of n, with a daily storage cost of $3430 dollars when 700 items are stored. In addition, there is a fixed daily cost of $7000. 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

Solution :

$T_1$ = k.n (cost proportional to quantity)

35000 = k . 700

k = $\frac{35000}{700}$

  = 50

So $T_1$ = 50 n

Now $T_2=Pn^2$     (cost proportional to square of n)

$3430= P(700)^2$

3430 = 490000 P

$P = \frac{3430}{490000}$

$P = \frac{7}{1000}$

$T_2=\frac{7}{1000}n^2$

Now, $T_3= 7000$   (fixed)

Therefore,

Total, $T=T_1+T_2+T_3$

$T=50n+\frac{7}{1000}n^2+7000$

$T=0.007n^2+50n+7000$

Answer 2

The required formula in terms of T and n is T = 0.000$n^{2}$ + 50 n +7000.

Given that ,

The total daily cost = T

When producing and storing n items is the sum of production, storage and fixed costs.

The manufacturing cost of a product when 700 items produced = $35000 dollars per day.

Daily storage cost of when 700 items are stored = $3430 dollars

We have write the formula in terms of total daily cost T and n stored items.

According to the question,

The cost to manufacture a product is proportional to the quantity produced, with a cost of $35000 dollars per day when 700 items are produced.

Where ,total daily cost T and n stored items.

$T_1 = k . n$

$35000 =  k.700

$k = \frac{35000}{700}$

k = $50

The cost to store n items is proportional to the square of n, with a daily storage cost of $3430 dollars when 700 items are stored.

$T_2 = pn^{2}$

$3430 = P $(700)^{2}$

$3430 = P 490000

$P = \frac{3430}{490000} \\\\P = \frac{7}{1000}$

There is a fixed daily cost of $7000.

$T_3 = 7000$

Combine all the equation ,

Where T = $T_1 = T_2 = T_3$ and p = k =n

T = 0.000$n^{2}$ + 50 n +7000

Hence,  The required formula in terms of T and n is T = 0.000$n^{2}$ + 50 n +7000.

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