If it costs $20 to set up and later clean a bagel press and bagels cost $1 per week per bagel to store, how many times should the bagel press be run each week to produce 1,000 bagels a week to be sold continuously?
A)Once
B)Twice
C)3 times
D)4 times
E)5 times

Question

Quizplus · Accepted Answer

Setting up and cleaning the bagel press costs a fixed $20, so we want to minimize the weekly cost of storing the bagels. If we make all 1,000 bagels at once, we will have to pay $1,000 each week to store them. However, if we make smaller batches, the cost of storage will be less since we only pay $1 per bagel per week. We want to find the number of batches that minimizes the cost of storage.

Let's assume we make x batches each week, with each batch consisting of n bagels. Then we have:

Total number of bagels made each week = x * n
Cost of storage each week = $1 per bagel * x * n = $x * n

We want to minimize the cost of storage subject to the constraint that we make 1,000 bagels each week. That is:

x * n = 1,000

Rearranging this equation, we get:

n = 1,000 / x

Substituting this into the equation for the cost of storage, we get:

Cost of storage = $x * n = $x * (1,000 / x) = $1,000/x

To find the value of x that minimizes this expression, we can take the derivative with respect to x and set it equal to zero:

d/dx (Cost of storage) = d/dx ($1,000/x) = -$1,000/x^2
-$1,000/x^2 = 0
x = sqrt(1000) = 31.6

Since we can't make a fractional number of batches, we should round up to the next integer, which is x = 32. Therefore, we should run the bagel press 32 times each week in batches of 1,000/32 = 31.25 ≈ 31 bagels. This will minimize the cost of storage to $32 per week.

If It Costs $20 to Set Up and Later Clean