A company offers the following schedule of charges: $30 per thousand for orders of 50,000 or less with the charge per thousand decreased by 37.5 cents for each thousand above 50,000. Find the order which will make the company's receipts a maximum?

How do you solve it?

right now:

cost per thousand = 30
orders = 50 000

for every increase of 1000 orders, cost decreases by .375

net the number of 1000 increases be n
cost per thousand = 30 - .275n
orders = 50,000 + 1000n

receipts = (30-.375n)(50000 + 1000n)
= 1,500,000 + 30,000n - 18750n - 375n^2
= -375n^2 + 11250n +1500000
d(receipts)/dn = -750n + 11250
= 0 for a max of receipts
750n = 11250
n = 15

for a max recepts
number should be 50000+1000(15) = 65000
cost per thousand = 30 - .375(15) = $24.375

testing:
for the above answer :
receipts = 65000(24.375) = 1, 584,375

take n = 14
number = 64000
cost = 30-.375(14) = $24.75
receipts = 64000(24.75) = 1, 584,000 , ahh a bit less

take n = 16
number = 66000
cost = 30-.375(16) = $24
receipts = 66000(14) = 1,584,000 , again a bit less

My answer is correct.

what is the final answer?

Well, let me first put on my thinking wig... I mean, hat.

To solve this problem, we need to find the order that will make the company's receipts maximum.

Let's break it down. If you order 50,000 or less, the charge is $30 per thousand. However, if you order more than 50,000, the charge decreases by 37.5 cents for each additional thousand.

So, let's say you place an order for x thousands. If x is less than or equal to 50, the cost will be 30x. But if x is greater than 50, we need to subtract 37.5 cents for each additional thousand. Since we know that one dollar is equal to 100 cents, we can calculate the charge for orders greater than 50,000 as (30 - 0.375x) * x.

To maximize the company's receipts, we need to find the value of x that will maximize the total charge.

Now, let me juggle some equations here...

To find the maximum, we can differentiate the total cost equation with respect to x, set it to zero, and solve for x. So, the derivative of the total cost function is 30 - 0.75x.

Setting the derivative equal to zero, we get 30 - 0.75x = 0.

Solving for x, we find that x = 40.

So, an order of 40,000 will make the company's receipts maximum. However, I must warn you, this is a mathematically optimal solution. It may not take into account other factors like supply and demand, market conditions, or the possibility of clowns performing free shows at your doorstep. So, take it with a pinch of circus dust!

To find the order that will make the company's receipts a maximum, we need to determine the charge per thousand and calculate the total receipts for different quantities. Here's how you can solve the problem step-by-step:

1. Determine the charge per thousand:
- For orders of 50,000 or less, the charge is $30 per thousand.
- For each thousand above 50,000, the charge per thousand decreases by 37.5 cents.

To express the charge per thousand as a function of the quantity, we can use the following equation:
C(q) = $30 - ($0.375 × (q - 50)), where C(q) represents the charge per thousand for a given quantity (q).

2. Calculate the total receipts for different quantities:
- The total receipts can be calculated by multiplying the charge per thousand by the quantity and then dividing it by 1,000.
- The formula for the total receipts (R) in terms of the quantity (q) is:
R(q) = (C(q) × q) / 1000.

3. Find the maximum receipts:
To find the order that will make the company's receipts a maximum, you can create a table or use a graphing calculator to calculate the total receipts for different quantities.
Start by selecting a range of quantities, such as 0 to 100,000, with increments of 1,000. Calculate and record the total receipts for each quantity using the formula R(q).

Once you have the complete table of quantities and corresponding receipts, identify the quantity that yields the maximum receipts. This will be the order that makes the company's receipts a maximum.

To solve this problem, we need to find the order which will maximize the company's receipts.

Let's break down the information given in the question:

1. The cost per thousand for orders of 50,000 or less is $30.
2. The charge per thousand decreases by 37.5 cents for each thousand above 50,000.

Now, let's approach the problem using some algebra.

Let's assume the number of thousands in the order is represented by x.

For orders of 50,000 or less (x <= 50), the cost per thousand is always $30. So, the total cost (C) for this range can be calculated as follows:
C = 30x

For orders above 50,000 (x > 50), the charge per thousand decreases by 37.5 cents for each thousand above 50,000. So, the cost per thousand (P) for this range can be calculated as follows:
P = 30 - (x - 50)(0.375)

To find the company's receipts (R), we need to calculate the total amount charged (T) for the given order:
T = Px
R = Tx

Now we have all the equations needed to find the order that will maximize the company's receipts.

1. For x <= 50:
C = 30x

2. For x > 50:
P = 30 - (x - 50)(0.375)
T = Px

To find the order which will make the company's receipts a maximum, we need to find the value of x that maximizes R.

We can do this by finding the value of x for which the derivative of R with respect to x is equal to zero. This will give us a critical point which can be a maximum or minimum.

Differentiating R with respect to x:
dR/dx = d(Tx)/dx
= T + x(dT/dx)

Setting dR/dx equal to zero:
0 = T + x(dT/dx)

Now, we can solve this equation to find the value of x that maximizes R.

Note: The exact calculations to find the maximum order size require numerical methods, such as calculus or optimization algorithms.