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Answer:
37,528 containers of cookies
Step-by-step explanation:
a) <u>Cost Function</u>
For the fixed annual demand of 500,000 containers, the manufacturing cost of $0.53 each will total ...
manufacturing cost = 500,000×$0.53 = $265,000 annually.
For a production batch size of x containers, those x containers will be put into storage, and withdrawn at the uniform rate of 500,000 containers per year. On average, (1/2)x containers will be in storage, so storage costs will be ...
storage cost = (1/2)($0.36x) = $0.18x . . . . annually
For annual demand of 500,000 containers, and a production batch size of x, there will be 500,000/x production batches each year. The setup cost is $507 for each of those, so the annual setup cost is ...
setup cost = (500,000/x)($507) = $253,500,000/x . . . . annually
The total cost of producing 500,000 containers annually will be ...
C = setup cost + storage cost + manufacturing cost
C(x) = 253,500,000/x + 0.18x + 265,000
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b) <u>Minimal-Cost Batch Size</u>
Cost will be minimized when its derivative with respect to x is zero.
dC/dx = -253,500,000/x^2 +0.18 = 0
x^2 = 253,500,000/0.18 . . . . . . . solve for x^2
x ≈ 37,527.8 ≈ 37,528 . . . . . . . . . take the square root
The size of the production run that will minimize production cost is 37,528 boxes.
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<em>Comment on the solution</em>
You may notice that the equation we finally solve for batch size is equivalent to one that sets setup cost equal to storage cost:
253,500,000/x = 0.18x
This relation (setup cost = storage cost) is the general solution for this sort of problem regarding batch size.
Answer:
The fraction of the area of ACIG represented by the shaped region is 7/18
Step-by-step explanation:
see the attached figure to better understand the problem
step 1
In the square ABED find the length side of the square
we know that
AB=BE=ED=AD
The area of s square is
where b is the length side of the square
we have
substitute
therefore
step 2
Find the area of ACIG
The area of rectangle ACIG is equal to
substitute the given values
step 3
Find the area of shaded rectangle DEHG
The area of rectangle DEHG is equal to
we have
substitute
step 4
Find the area of shaded rectangle BCFE
The area of rectangle BCFE is equal to
we have
substitute
step 5
sum the shaded areas
step 6
Divide the area of of the shaded region by the area of ACIG
Simplify
Divide by 5 both numerator and denominator
therefore
The fraction of the area of ACIG represented by the shaped region is 7/18
