Need help to solve this word problem

Question

Demand and cost equation.

q = quantity in kg

p = price in $

c = total costs in $

p = 350 - 68q

c = 265 + 70q

a. Break-even level(s) of production to four decimal places.

b. Profit maximising level of production to four decimal places.

I have given a. a go and came to the answers q1 = 1474 and q2 = 2643 . But I am not sure that they are correct.

I determined that the variable cost = p(350), the fixed cost = c(265) and of course q = to kg.

Answer 1

Something did seem odd with this problem until I realized the quantities were in kilograms and that we might be able to sell the products in fractions of a kilogram--as indicated in that the request for the answer to be accurate to 1/10,000 of a kilogram. I reached numbers similar to Peter--more specifically, 1.4743 kg and 2.6434 kg--derived from the following equation:

70 / 34 +/- square root (395) / 34 kg

These numbers would also have matched your results except your numbers show the number of grams instead of the number of kilograms shown to the 4th decimal place.

Also, I do not understand what you are referring to when you said the variable costs = p(350).

If Total Cost (TC) = c = 265 + 70q ==> variable cost = 70

But break even is determined by setting TR = TC where TR is Total Revenue and TC is Total Cost--and TR = p * q:

TR = pq = 350q - 68q²

TC = 265 + 70q = 350q - 68q² = TR

After adding 68q^2 -350q to each side you get:

68q^2 -280q +265 = 0

Using the quadratic formula, you get:

280 +/- square root(78400 - 4 (68) (265))

q = --------------------------------------------------------

136

Now those look like some pretty ugly number, but you can simplify it a bit and end up with:

70 +/- square root(395)

q = -----------------------------------

34

q = 1.47428 or q = 2.64337

Since the Break-even level refers to the first quantity that will yield a profit, and the quantity was to be shown with 4 decimal places, the answer is:

q = 1.4743 kg

A good way to check your work is to build a table:

| q | p | TR | TC | Profit |

| 1.4742 | 249.754 | 368.188 | 368.194 | (0.006) |

| 1.4743 | 249.747 | 368.203 | 368.201 | 0.002 | <<<

| 1.4744 | 249.740 | 368.218 | 368.208 | 0.010 |

The table supports the above calculated results and shows that the first quantity that will generate a profit is:

q = 1.4743

In order to demonstrate that there was a profit, I showed the dollars with 3 decimal places. Also, if the quantities were sold in grams, one would need to round up to 1475 g and, if sold only in kilograms, one would need to round up to 2 kg.