A manufacturer of a certain brand of appliance estimates that he can sell 5000 units a year at Php 900.00 each and that he can sell 1500 units more per year for each Php 100.00 decrease in price? What price per unit will give the greatest returns?
Let x be the number of price decreases. Then the revenue is
r(x) = (900-100x)(5000+1500x)
= 100*500(9-x)(10+3x)
= 50000(-3x^2+17x+90)
Now find when dr/dx=0 and you can figure the price for maximum revenue.
let the number of 100 decreases be n
number sold = 5000 + 1500n
selling price per item = 900 - 100n
return = (5000+1500n)(900-100n)
= 500(10 + 3n)(100)(9 - n)
= 50000(90 + 17n - 3n^2)
d(return)/dn = 50000(17 - 6n)
= 0 for a max
6n = 17
n = 17/6 = appr 2.8
assuming we can only decrease in multiples of 100
for 2 decreases:
number = 8000
price = 700
return = 5 600 000
for 3 decreases:
number = 9500
price = 600
return = 5 700 000
there should be 3 decreases of 100
To find the price per unit that will give the greatest returns, we need to determine the price point where the total revenue is maximized.
Let's break down the given information:
- The manufacturer estimates that 5000 units can be sold per year at a price of Php 900.00 each.
- For each Php 100.00 decrease in price, an additional 1500 units can be sold per year.
To determine the price per unit that will maximize returns, we need to analyze the total revenue for various price points. Let's calculate this step by step:
1. Determine the number of units sold at the current price of Php 900.00:
Total Units Sold = 5000
2. Calculate the revenue generated at the current price:
Total Revenue at Php 900.00 = 5000 units * Php 900.00 per unit
3. Determine the price decrease necessary to sell an additional 1500 units:
Change in units sold = 1500
Change in price = Php 100.00
Price decrease required = Change in units sold * Change in price
4. Calculate the new price per unit:
New price per unit = Current price per unit - Price decrease required
5. Calculate the additional revenue generated from selling the additional units at the new price:
Additional Revenue = Additional units * New price per unit
6. Determine the total revenue at the new price:
Total Revenue at the new price = Total Revenue at the current price + Additional Revenue
7. Repeat steps 4-6 with successive price decreases until you reach a point where decreasing the price further does not result in a significant increase in revenue.
8. Compare the total revenue at each price point and identify the price per unit that gives the greatest returns.
Note: Since the specific values for the price decrease required and additional units sold are not provided, we cannot precisely calculate the price per unit that will give the greatest returns. We need additional information to provide a more accurate solution.
To determine the price per unit that will yield the greatest returns, we need to calculate the revenue for each price and find the maximum value.
Let's start by creating a table to find the revenue at different prices:
Price per unit (Php) | Quantity sold | Revenue (Php)
--------------------------------------------------
900 | 5000 | 900 * 5000
800 | 6500 | 800 * 6500
700 | 8000 | 700 * 8000
600 | 9500 | 600 * 9500
500 | 11000 | 500 * 11000
To find the revenue at each price, we multiply the price per unit by the quantity sold.
Now, let's calculate the revenue for each price:
Revenue at 900 Php = 900 * 5000 = 4,500,000 Php
Revenue at 800 Php = 800 * 6500 = 5,200,000 Php
Revenue at 700 Php = 700 * 8000 = 5,600,000 Php
Revenue at 600 Php = 600 * 9500 = 5,700,000 Php
Revenue at 500 Php = 500 * 11000 = 5,500,000 Php
From the table, we can see that the greatest revenue is at a price of 600 Php per unit, which yields 5,700,000 Php.
Therefore, the price per unit that will give the greatest returns is 600 Php.