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?
if there are x decreases of 100, then
price = 900-100x
demand = 5000+1500x
so, the revenue r is
r = (900-100x)(5000+1500x)
So, just find where dr/dx = 0 for the maximum revenue. You know it is a maximum since the parabola opens downward.
Note that x is not the answer to the question.
so how do we find the answer?i still don't get it
Every two years, a manufacturer's total yearly sales decline 20%. She sells 400 total units in her first year. Rounded to the nearest unit, how many total units will she sell her ninth year?
To find the price per unit that will give the greatest returns, we need to analyze the relationship between price and number of units sold.
Let's say the price per unit is P (in Php) and the number of units sold per year is N. According to the given information, the following relationship holds:
N = 5000 + (P - 900) / 100 * 1500
This equation represents the additional units sold for each Php 100.00 decrease in price.
To calculate the total revenue, we multiply the number of units sold by the price per unit:
Revenue = P * N
To find the price that will give the greatest returns, we can create a table to examine the relationship between price and revenue.
Let's start with a price of Php 1000:
N = 5000 + (1000 - 900) / 100 * 1500 = 6500
Revenue = 1000 * 6500 = Php 6,500,000
Now, let's calculate the revenue for different prices, decreasing by Php 100 each time:
Price (P) | Units Sold (N) | Revenue (P * N)
1000 | 6500 | Php 6,500,000
900 | 8000 | Php 7,200,000
800 | 9500 | Php 7,600,000
700 | 11000 | Php 7,700,000
600 | 12500 | Php 7,500,000
500 | 14000 | Php 7,000,000
400 | 15500 | Php 6,200,000
300 | 17000 | Php 5,100,000
From the table, we observe that as the price decreases, the number of units sold increases. However, there's a trade-off because the reduction in price leads to a decrease in revenue.
The price per unit that will give the greatest returns is when the revenue is at its highest value. In this case, the greatest revenue is Php 7,700,000, which occurs when the price per unit is Php 700.
Therefore, the price per unit that will give the greatest returns is Php 700.