SOLUTION: Maximum profit: A kitchen appliance manufacturer can produce up to 200 appliances per day. The profit made from the sale of these machines can be modeled by the function P(x)=-0.5x

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Question 1128209: Maximum profit: A kitchen appliance manufacturer can produce up to 200 appliances per day. The profit made from the sale of these machines can be modeled by the function P(x)=-0.5x^2+175x-3300 where P(x) is the profit in dollars, and x is the number of appliances made and sold.
a)Find the y intercept and explain what it means in this context.
b)Find the x intercepts and explain what they mean in this context.
c)Determine the domain of the function and explain its significance.
d)How many should be sold to maximize profit? What is the maximum profit?
Found 2 solutions by Shin123, MathTherapy:
Answer by Shin123(626) About Me  (Show Source):
You can put this solution on YOUR website!
a) The y intercept occurs when x=0.
Solved by pluggable solver: Completing the Square for Quadratics
To complete the square for the quadratic -0.5%2Ax%5E2%2B175%2Ax-3300=0, we must first find a square which when expanded, has -0.5x2 and 175x in it.
Factoring -0.5 from the left side gives -0.5%28x%5E2-350%2Ax%2B6600%29=0. %28x-175%29%5E2 is the square we are looking for. So we get -0.5%28%28x-175%29%5E2-24025%29=0. Taking the -24025 out of the -0.5, we get highlight%28-0.5%28x-175%29%5E2%2B12012.5%29. Subtracting 12012.5 from both sides, we get -0.5%28%28x-175%29%5E2%29=-12012.5. Since the right side is negative, there are no real solutions.
Plugging in x=0 gives y=-3300. The y intercept shows the profit if they make no appliances.
b) From the above, the x-intercepts are x=330 and x=20. The x-intercepts show where they make no profit.
c) -0.5%28x-175%29%5E2%2B12012.5=0 Since the square of any real number is nonnegative and the square is multiplied by a negative number, the maximum profit occurs when %28x-175%29%5E2=0. This occurs when x=175. They make 12012.50 dollars if they make 175 appliances.

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Maximum profit: A kitchen appliance manufacturer can produce up to 200 appliances per day. The profit made from the sale of these machines can be modeled by the function P(x)=-0.5x^2+175x-3300 where P(x) is the profit in dollars, and x is the number of appliances made and sold.
a)Find the y intercept and explain what it means in this context.
b)Find the x intercepts and explain what they mean in this context.
c)Determine the domain of the function and explain its significance.
d)How many should be sold to maximize profit? What is the maximum profit?
To find the x-intercepts, you don't have to take the COMPLEX approach as the other person did.

Profit equation: 

(x - 330)(x - 20) = 0

As seen above, the x-intercepts are: highlight_green%28matrix%281%2C3%2C+330%2C+and%2C+20%29%29

d1) Also, maximum units for maximum profit is realized at the point where matrix%281%2C3%2C+x%2C+%22=%22%2C+-+b%2F%282a%29%29, or where: matrix%281%2C5%2C+x%2C+%22=%22%2C+-+175%2F%282+%2A+-+.5%29%2C+%22=%22%2C+175%29

d2) Maximum profit occurs at: