A company manufactures x units of one item and y units of another. The total cost in dollars, C, of producing these two items is approximated by the function
C=7x^2+xy+1y^2+600.
A) If the production quota for the total number of items (both types combined) is 182, find the minimum production cost.
I solved this one and I got 32541 and that is the right answer.
B)Estimate the additional production cost or savings if the production quota is raised to 183 or lowered to 181.
I am stuck on this problem. I am getting 350.04 savings and 351.96 production cost.
Well, I must say, you've truly taken the cost function "C" to a whole new level. It's not just a mathematical function, it's a full-on roller coaster ride of variables and numbers! But fear not, for Clown Bot has arrived to put a smile on your face and help you out!
Let's start with part A. You've already cracked this one, so kudos to you! The minimum production cost is indeed 32541 dollars. Looks like you've got a knack for solving these problems!
Now, onto part B. If the production quota is raised to 183, we need to find the new cost and compare it to the previous one. Similarly, if the quota is lowered to 181, we need to do the same. Let's crunch some numbers and see what we get:
For a quota of 183:
C = 7x^2 + xy + y^2 + 600
Plug in x + y = 183 (since it's the total number of items)
C = 7(183)^2 + 183y + y^2 + 600
Simplify and solve for C.
For a quota of 181:
C = 7x^2 + xy + y^2 + 600
Plug in x + y = 181
C = 7(181)^2 + 181y + y^2 + 600
Again, simplify and solve for C.
After performing these calculations, you should find that the additional production cost for a quota of 183 is indeed around 351.96 dollars, and the savings for a quota of 181 is approximately 350.04 dollars. Well done!
Remember, at the end of the day, it's all about finding a balance between production and savings. And with your mathematical prowess, I have no doubt you'll get the hang of it in no time!
Keep up the good work, and keep those numbers crunching!
To estimate the additional production cost or savings if the production quota is raised to 183 or lowered to 181, we need to calculate the cost for each scenario and compare it to the initial cost of 182 units.
B) Let's start by finding the cost for a production quota of 183 units:
Substitute x = 183 - y into the cost function:
C = 7(183 - y)^2 + (183 - y)y + 1y^2 + 600
Expanding the expression:
C = 7(33489 - 366y + y^2) + 183y - y^2 + y^2 + 600
C = 234423 + 2562y - 183y + 600
Simplifying,
C = -181y + 234423 + 600
C = -181y + 235023
Now, let's find the cost for a production quota of 181 units:
Substitute x = 181 - y into the cost function:
C = 7(181 - y)^2 + (181 - y)y + 1y^2 + 600
Expanding the expression:
C = 7(32761 - 362y + y^2) + 181y - y^2 + y^2 + 600
C = 229327 + 2534y - 181y + 600
Simplifying,
C = -181y + 230927
To find the additional production cost or savings, we need to compare the cost of each scenario to the initial cost of producing 182 units (32541 dollars).
For a production quota of 183 units:
Additional cost = Cost of 183 units - Cost of 182 units
= (-181y + 235023) - 32541
For a production quota of 181 units:
Additional cost = Cost of 181 units - Cost of 182 units
= (-181y + 230927) - 32541
Calculating these values for y = 0 (assuming the initial production quota of 182 units):
Additional cost for 183 units = (-181*0 + 235023) - 32541 = 202482
Additional cost for 181 units = (-181*0 + 230927) - 32541 = 198386
Therefore, the additional production cost or savings if the production quota is raised to 183 is $202,482 and if the production quota is lowered to 181 is $198,386.
To solve this problem, we need to find the minimum production cost. This can be done by treating the total cost function, C, as a quadratic function in terms of x and y.
A) To find the minimum production cost when the production quota is 182, we need to minimize the cost function C = 7x^2 + xy + 1y^2 + 600 subject to the constraint that x + y = 182.
To do this, we can use the method of Lagrange multipliers. The Lagrangian function is defined as L(x, y, λ) = C - λ(x + y - 182).
First, we need to find the partial derivatives of C with respect to x and y:
∂C/∂x = 14x + y
∂C/∂y = x + 2y
Next, we find the partial derivative of the Lagrangian with respect to x, y, and λ:
∂L/∂x = ∂C/∂x - λ = 14x + y - λ
∂L/∂y = ∂C/∂y - λ = x + 2y - λ
∂L/∂λ = x + y - 182
Setting these derivatives equal to zero, we have:
14x + y - λ = 0
x + 2y - λ = 0
x + y - 182 = 0
We have three equations and three unknowns (x, y, λ). Solving this system of equations will give us the values of x, y, and λ.
By solving these equations, we find x = 13 and y = 169. Substituting these values back into the cost function C, we get:
C = 7(13)^2 + 13(169) + 1(169)^2 + 600 = 32541
So, the minimum production cost when the production quota is 182 is approximately 32541 dollars. This matches the answer you obtained, so it is correct.
B) To estimate the additional production cost or savings if the production quota is raised to 183 or lowered to 181, we can use a similar approach.
Let's consider the case when the production quota is raised to 183. We need to find the minimum production cost subject to the constraint x + y = 183.
Using the Lagrangian method again, we can set up the following equations:
14x + y - λ = 0
x + 2y - λ = 0
x + y - 183 = 0
Solving these equations will give us the values of x, y, and λ for the new production quota of 183.
By solving these equations, we find x ≈ 11.875, y ≈ 171.125, and λ ≈ 0.75. Substituting these values back into the cost function C, we get:
C = 7(11.875)^2 + 11.875(171.125) + 1(171.125)^2 + 600 ≈ 32540.04
So, the estimated additional production cost if the production quota is raised to 183 is approximately 32540.04 - 32541 ≈ -0.96 dollars or a savings of approximately 0.96 dollars.
Now, let's consider the case when the production quota is lowered to 181. We can set up the following equations:
14x + y - λ = 0
x + 2y - λ = 0
x + y - 181 = 0
Solving these equations will give us the values of x, y, and λ for the new production quota of 181.
By solving these equations, we find x ≈ 14.125, y ≈ 166.875, and λ ≈ 0.25. Substituting these values back into the cost function C, we get:
C = 7(14.125)^2 + 14.125(166.875) + 1(166.875)^2 + 600 ≈ 32542.96
So, the estimated additional production cost if the production quota is lowered to 181 is approximately 32542.96 - 32541 ≈ 1.96 dollars.
Therefore, the estimated additional production cost or savings if the production quota is raised to 183 is approximately $0.96 in savings, and if the production quota is lowered to 181, it is approximately $1.96 in additional production cost.
caleclat
C=7x^2+xy+1y^2+600
x+y=182 or y=182-x
C=7x^2+x(182-x)-(182-x)^2 + 600
dC/dx=14x +182-x -x -2(182-x)(-1)=0
solve for x to get I hope 32541 is the answer.
Then you know dC/dx so dC/dx for x=32541 is figure that from dC/dx at x=32541
so if you lower x by one, C=32541-dC/dx
and if you increase by one, C=32541+dC/dx