Expert: Scotto - 3/20/2009

Question
1. For the given cost function
C(x)=5300+340 x + 1.7 x^2 and the demand function p(x) = 1020.
Find the production level that will maximize profit.

2. The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 300 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 2 dollar increase in rent. Similarly, one additional unit will be occupied for each 2 dollar decrease in rent. What rent should the manager charge to maximize revenue?

3. The cost in dollars of producing x units of a product is given by
C(x) = \frac{6 x^2 - 21 x + 3}{3 \sqrt{x}} for x \ge 0.
Determine the value of x when the marginal cost is 0.

The number of units is


THANKS IN ADVANCE


Answer
1. For the given cost function
C(x)=5300+340 x + 1.7 x^2 and the demand function p(x) = 1020.
Find the production level that will maximize profit.

Take the derivative, set it to 0, and this will be the value for x.
We know that C'(x) is 340 + 3.4x, so take 340 + 3.4x = 0 and solve for x.  This will gives us x is -100.  This sounds like the problem was entered wrong, since x is usually a quantity or size, which can't be a negative number.


2. The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 300 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 2 dollar increase in rent. Similarly, one additional unit will be occupied for each 2 dollar decrease in rent. What rent should the manager charge to maximize revenue?

1 increase in 100 implies 2 decrease in 300.
1 decrease in 100 implies 2 increase in 300.
With these two notes, the equation would be R(x)=(100+x)(300-2x).
To maximize this, we'll have to look at R'(x).
Lets multiply out the quanties first getting 30,000 + 100x - 2x².
If this is R(x), then R'(x) = 100 - 4x.
Setting this equal to zero gives us 100 = 4x, so x = 25.

The revenue R(25) = 125 * 25 = 5^3 * 5^2 = 5^5 = 3,125.
Or you could just think that the value is 125*20 + 125*5 =
250(0) + 625 = 3,125.  Then again, most people just use a calculator.


3. The cost in dollars of producing x units of a product is given by
C(x) = frac(6x² - 21x + 3)(3/√x) for x ≥ 0 – is this correct?
Determine the value of x when the marginal cost is 0.

The number of units is – I don’t know.  Let’s find out.

The marginal cost would be C’(x), where C(x) = frac(6x² - 21x + 3)(3/√x).

The frac(6x² - 21x + 3) is 0 wherever x is 0.  This occurs whenever x is an integer and at the solutions to the equation 6x² - 21x + 3 = 3(2x² - 7x + 1) = 0.  That’s where x is (7 ±(49-8))/4 = (7±√41)/4.

When the function is not near one of those values, we can treat C(x) like there is no fraction term when finding C’(x).  Note that this does not change C(x), but we just trying to find C’(x).  Doing so gives us that C(x) = (18x² - 63x+9)/√x, which is 18√x^3 – 63√x + 9/√x.  This would make C’(x) = 27√x – 31.5/√x – 4.6/√x^3.  Factoring out √x gives C’(x) = √x(27 – 31.5/x +4.6/x²).  If we remake the front into √x/x², our new equation is C(x) = (√x/x²)(27x² - 31.5x + 4.6).  Using the quadratic, we get x = (31.5±√(992.25-124.2))/54 = (31.5±√868.05)/54.  This would give us values where the function levels out.  However, we also need to check the integers and see what the value is near them.  I would do so with Excel.