Your firm offers to deliver 250 tables to a dealer, at $160 per table, and to reduce the price per table on the entire order by 50 cents for each additional table over 250.
Find the dollar total involved in the largest possible transaction between the manufacturer and the dealer; then find the smallest possible dollar amount.
cost of x tables
= [160-.5(x-250)]x for x>/=250
c = 160 x + 125 x - .5 x^2
c = 285 x - .5 x^2
dc/dx = 285 - x
max when x = 285
then c = .5*285^2 = 40,612.50
and c = 0 when x = 0 of course but also when x = 285*2 = 570
Well, if we're talking about the largest possible transaction, I would suggest delivering 1 million tables. That way, the dealer would be getting a discount of 50 cents for each additional table, which would come out to a whopping $500,000 discount off the original price of $160 per table.
So, the largest possible transaction would be for 1 million tables, resulting in a total discount of $500,000,000. That's one heck of a deal!
Now, for the smallest possible dollar amount, I guess we could deliver zero tables. That would give the dealer exactly zero dollars in total.
But hey, that's no fun! Let's focus on the million-table extravaganza instead!
To find the dollar total involved in the largest possible transaction, we need to calculate the price per table for 250 tables and then subtract 50 cents for each additional table.
Step 1: Calculate the price per table for 250 tables.
Price per table = $160
Therefore, the total cost for 250 tables = 250 * $160 = $40,000
Step 2: Calculate the reduction in price per table for each additional table over 250.
Reduction in price per additional table = $0.50
Step 3: Calculate the number of additional tables beyond 250.
Number of additional tables = Total number of tables - 250
Step 4: Calculate the reduction in total cost due to the additional tables.
Reduction in total cost = Number of additional tables * Reduction in price per additional table
Step 5: Calculate the total cost for the largest possible transaction.
Total cost = Total cost for 250 tables - Reduction in total cost
Let's assume the total number of tables in the largest possible transaction is 300.
Number of additional tables = 300 - 250 = 50
Reduction in total cost = 50 * $0.50 = $25
Total cost = $40,000 - $25 = $39,975
Therefore, the dollar total involved in the largest possible transaction between the manufacturer and the dealer is $39,975.
To find the smallest possible dollar amount, we need to calculate the total cost for the minimum number of tables.
Let's assume the total number of tables in the smallest possible transaction is 250.
Number of additional tables = 250 - 250 = 0
Reduction in total cost = 0 * $0.50 = $0
Total cost = $40,000 - $0 = $40,000
Therefore, the smallest possible dollar amount involved in the transaction is $40,000.
To find the largest possible transaction, we need to determine how many tables the dealer would need to purchase to maximize the discount and subsequently minimize the price per table.
Let's start by calculating the number of tables required to minimize the price per table. Given that the price per table decreases by 50 cents for each additional table over 250, we can set up an equation:
Price per table after discount = $160 - $0.50 * (Total tables - 250)
To minimize the price per table, we want to find the point where the derivative of this equation is equal to zero. Taking the derivative and setting it equal to zero:
d/dx (Price per table after discount) = -0.50 = 0
Solving for x, we find:
-0.50 = 0
x = 250
Therefore, the dealer should purchase a minimum of 250 tables to get the lowest price per table possible.
To calculate the largest possible transaction, we need to determine the total cost when the dealer purchases the minimum required tables (250 tables) and then add any additional tables beyond that at the discounted rate.
For the minimum requirement, the total cost would be:
Total cost = Price per table * Number of tables
Total cost = $160 * 250
Total cost = $40,000
Now, let's consider additional tables beyond the minimum requirement. Since the price per table decreases by 50 cents for each additional table, we can calculate the total cost by summing up the cost for each additional table:
Additional tables = Total tables - 250
Total cost for additional tables = (Price per table after discount - $160) * Additional tables
Total cost for additional tables = ($160 - $0.50 * (Total tables - 250)) * (Total tables - 250)
To find the maximum possible transaction, we need to determine the total cost when the maximum number of additional tables is purchased. We can do this by setting the derivative of the total cost equation equal to zero and solving for the number of additional tables:
d/dx (Total cost for additional tables) = 0
Solving for x, we find:
-0.50 * (Total tables - 250) - 0.50 = 0
Total tables - 250 = 1
Total tables = 251
So, the maximum number of additional tables that should be purchased to maximize the discount is 251.
Substituting this value for Total tables in the total cost equation:
Total cost for additional tables = ($160 - $0.50 * (251 - 250)) * (251 - 250)
Total cost for additional tables = ($160 - $0.50) * 1
Total cost for additional tables = $159.50
Adding this cost for additional tables to the minimum cost, the total maximum transaction would be:
Total maximum transaction = Total cost for minimum requirement + Total cost for additional tables
Total maximum transaction = $40,000 + $159.50
Total maximum transaction = $40,159.50
Therefore, the largest possible transaction between the manufacturer and the dealer is $40,159.50.
To find the smallest possible dollar amount, we assume that the dealer only purchases the minimum required tables, as any reduction in the number of tables would increase the price per table.
Therefore, the smallest possible transaction is the total cost for the minimum requirement:
Total smallest transaction = Total cost for minimum requirement
Total smallest transaction = $40,000
Thus, the smallest possible dollar amount in the transaction between the manufacturer and the dealer is $40,000.