Question

Calculus Experts Help! Desperately need help. Thank you
"Your company titled BANGALORES R US produces bangalore torpedoes (commonly called bangalores,) a demolition used for breaching wire obstacles and mines. It produces these devices for the Army Corps of Engineers according to the production function Q = 12L+ 29L2-1.1L3, where Q is the number of bangalores produced per year and L is the number of laborers used per year for production. Additionally, the price of labor and other consumable materials (PL) is $7000 per laborer year. Each bangalore sells for $125.

Answer 1

For all functions "L" should be considered the independent variable.

Answer the following questions:
1.What is the minimum and maximum number of laborers possible for the production of Bangalores? This is the domain of what function?
2.How many laborers would be needed to maximize the production of bangalores?
3.How many laborers would be needed to maximize the Revenue for producing bangalores? What is that maximum revenue?
4.Is there a range of values of production for the bangalores that would be profitable? If so, what is this range? Include a graph to justify your answer.
5.Would your answer, from number 3, also give the maximum profit for producing bangalores? Why or why not?
6.What is the maximum profit that can be obtained by producing the bangalores? Show why this is a maximum using a graphical analysis of the first and second derivatives.
7.Economists refer to marginal revenue (MR) as the change in total revenue attributable to a one-unit change in output. The marginal cost (MC) is the change in total cost attributable to a one-unit change in output. In other words, marginal revenue is directly related to the derivative of the total revenue function and marginal cost is directly related to the derivative of the total cost function. Using your work above, explain why economists use the equation MR = MC to maximize profits.
8."The law of diminishing returns states that if increasing amounts of a variable factor (in this case, laborers) are applied to a production line, eventually a situation will be reached in which each additional unit of the variable factor adds less to the total production than the previous unit." Explain where the point of diminishing returns for the laborers is and why. What is a physical explanation in your company for the point of diminishing returns?"

Answer 2

I really need help. Any would be great! I can learn from the work shown too. Thanks!!! My first thought is to find the derivative of Q?

Answer 3

I tried. I need someone who is good at calculus. Someone help this desperate soul!

Answer 4

I would plot the function, just to see what it is doing
http://www.wolframalpha.com/input/?i=plot+q+%3D+12L%2B+29L%5E2-1.1L%5E3

to find the peak, take the derivative and set it equal to zero, and solve for L

Answer 5

for the first part

$$Q=12L+29L^2-1.1L^3=L(12+29L-1.1L^2)$$
by factoring furthur, we get,
$$Q=-1.1L\left(L^2-\frac{29}{1.1}L-\frac{12}{1.1}\right)=-1.1L\left(L-26.771\right)(L+0.407)$$
since it is clear that Q>=0,
$$L(26.771-L)(L+0.407)\geq0$$
$$L\geq0$$ therefore the minimum number of employees is 0,
if L>=0,L+0.407>0
$$(26.771-L)(L+0.407)\geq0 \implies L\leq26.771 $$
therefore maximum number of employees possible are 26

Answer 6

Are you sure? Wouldn't the derivative set to zero show that the slope is peaked, therefore a maximum? Why this way? isn't what you provided just a minimum? For example, the zeroes for the Y value on a graph?

Answer 7

Sorry for so many questions :)

Answer 8

For the first question you don't need calculus, I think you can find the maximum and minimum possible employees in the factory by my previous method or looking at the graph that phi has mentioned.

Answer 9

I appreciate your efforts. However, with the derivative set to zero, it comes in the form of ax^2 + bx + c = 0

Using the quadratic equation it comes to 17.78 giving it a local maximum. I think this must be correct ?

However I am not so sure about the other parts.

Answer 10

Yes. by differentiating Q wrt L you get the local maximum which gives the answer to the "Second" question

Answer 11

Now having found the number of laborers need to maximize the revenue (ie 17.78), find max[Q(17),Q(18)] to find the maximum revenue since a fractional number for L is not possible

Answer 12

Thank you for your time. I believe you are correct! :)
The function Q is the production of the Bangalores. Although the function extends past the maximum value from your method, they are however negative. This is illogical for the number of employees. The maximum is not the maximum for production, this is where I was confused. It is the maximum number of POSSIBLE employees. Thank you ! :)

Answer 13

So for number 3 I figured it is Q(18) x $125, because at near 18 laborers it is the maximum production.

However, shouldn't the cost of labor be considered? $7,000 per laborer? we have 18 laborers, so 18 x $7,000 = $126000

so for the total revenue, it should be [ Q(18) x $125 ] - 126,000

right?

Answer 14

You actually took the time to type up that question?

Answer 15

Hello, would you like to provide your perspective? :)

Answer 16

I think the cost for the labor is related to the "profit". Revenue is the income so the costs shoudl not be considered