A manufacturer can produce x units per week. The cost to produce these items is given by y=50x+20,000. These items will set at a unit price given by p=200-0.01x. How many items should the manufacturer produce in order to maximize profit? (Remember: revenue = number of units x unit price, and profit = revenue - total cost) (Related Rates)
Stacey Warren - Expert brainly.com
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So, the basic approach is to combine the functions you're given into a single profit equation, then find where the derivative = 0 to find a maximum (I'm assuming this is for a calculus class?)
So profit = revenue - cost
= x(200 - 0.01x) - 50x + 20000
= 200x - 0.01x^2 - 50x + 20000
= -0.01x^2 + 150x + 20000
The first derivative of that is:
d/dx (profit) = -.02x +50
and that will equal zero when x = 2500
Oh, woops...the derivative is wrong
the constant should be 150, which will make the final answer 7500, I believe
x stands for number of units
p stands for unit price
y stands for total cost
profit = number of units x unit price - total cost
profit = xp - y
profit = x(200-0.01x) - (50x+20000)
now just expand and collect like terms,
you should end up with a simplier quadratic equation.
To find the max profit, it depends on what you are learning.
I believe the most advanced way is using the derivative.
Another way is to find the vertex of the quadratic.