Question

a manufacturer can produce a color pen at a cost of $3. the color pens have been selling for $5 per pen and at this price, consumers have been buying 4000 pens per month. the manufacturer is planning to raise the price of the pens and estimates that for each $1 increase in price, 400 fewer pens will be sold each month. at what price should the manufacturer sell the pen to maximize profit? what is the maximum profit?

Answer 1

Let x be the number of increases of $1 a month.
Let y be the maximum profit.
The profit before the price increases:
$5-$3=$2
The profit when increasing the price:
y= (2+x x 1)(4000-400x)
y= 8000-800x +4000x -400x2 (x2: x square)
The vertex:
x= -3200/(-400 x 2) = 4
=> y= 14400
The price can maximize the profit is:
p= 3+ 2 +4x1= 9
The price can maximize the profit is $9
The maximum profit is $14400.