Information to solve problems
Q=12L+29L^2-1.1L^3
Q= # of cars produced/year
L=# of laborers used per year for production
Each car sells for $125.
$7000 per laborer per year is cost for labor and other consumable materials.
I attempted some parts of these questions and some parts I did not understand. Can someone check my work and help me with the parts I didn't get?
Questions:
1. What is the max and min number of laborers possible for the production of cars? This is the domain of what function?
I answered this one this way.
12L+29L^2-1.1L^3=0
L(12+29L-1.1L^2)=0
L=0, 26.77, -0.41
The minimum number of laborers would be 0.
The maximum number of laborers would be 26.77.
I don't know the answer to: this is the domain of what function?????
2. How many laborers would be needed to maximize the production of cars?
Q'=12+58L-3.3L^2
12+58L-3.3L^2=0
L=17.78, -0.20
Q'(max)=17.78
To maximize the production they need 17.78 laborers.
3. How many laborers would be needed to maximize the revenue for producing cars? What is the maximum revenue?
Revenue = price * production
=$125 * 26.8 = 3350 laborers ( I'm not sure if this is right)
Revenue (max)= price * Q'(max)
$125 * ((12*17.78) +(29*(17.78^2))-(1.1*(17.78^3))
Revenue(max)=$399,780.54
4. Is there a range of values of production for the cars that would be profitable? IF so, what is the range?
Profit = price*production-7000L
L(-137.5L^2+3625L-5500)=0
L=0, 1.56, 24.9
The range would be 1.56 - 24.9. (I'm not sure if this is right).
How would I justify this answer with a graph?
1. The function in question is the production function, which is given by Q = 12L + 29L^2 - 1.1L^3. The domain of this function represents the range of possible values for the number of laborers (L) used in the production of cars.
Your answer for the minimum and maximum number of laborers is correct. The minimum is 0, which means there could be no laborers involved in the production. The maximum is approximately 26.77 laborers.
To justify this answer with a graph, you can plot the production function on a graph with the number of laborers (L) on the x-axis and the quantity of cars produced (Q) on the y-axis. The points where the graph intersects the x-axis represent the minimum and maximum number of laborers, indicating the domain of the function.
2. To find the number of laborers needed to maximize the production of cars, you need to find the critical points of the production function. This can be done by finding the derivative of the function with respect to L and setting it equal to zero.
Differentiating the production function, Q'= 12 + 58L - 3.3L^2, and setting it equal to zero, we get:
12 + 58L - 3.3L^2 = 0
Solving this quadratic equation will give you the critical points, which represent the number of laborers that maximize production. In this case, you correctly found L = 17.78 as the value for maximizing production.
3. To maximize revenue, you need to consider the relationship between price (P), quantity (Q), and laborers (L). Revenue (R) is given by the equation R = P * Q.
You correctly calculated the Revenue (max) by plugging in the value Q'(max) = 17.78 (from the previous question) into the production function and multiplying it by the price per car ($125). The result of $399,780.54 is the maximum revenue.
4. To determine the profitable range of values for production, you need to consider the cost of production. Profit (P) is calculated as P = Revenue - Cost.
In this case, the cost per laborer per year is $7000. So, the profit equation becomes:
Profit = price (P) * production (Q) - cost (7000 L)
To find the range of values for production that would be profitable, you need to solve the quadratic equation Profit = 125Q - 7000L - 137.5L^2 + 3625L - 5500 = 0.
You obtained the correct critical points L = 1.56 and L = 24.9.
To justify this answer with a graph, plot the profit function on a graph with the quantity of cars produced (Q) on the x-axis and the profit (P) on the y-axis. The points where the graph intersects the x-axis represent the range of values for production that would be profitable. You can use these points to determine the range, which is approximately 1.56 to 24.9 cars produced.
Remember, it's always a good idea to double-check your calculations and ask for help if there are any uncertainties.