Question 473563
Suppose you own a strip mall. You want to determine the amount of rent you should charge to maximize your profit. The given equation measure the profit, P (is in thousands of dollars), per square foot of rental space (m is dollars per square foot). 
P(d) = -8.3m^2 + 53.1m - 26.5
{{{graph(400,400,-10,50,-10,150,-8.3x^2 + 53.1x - 26.5)}}}
  
1. Does the graph of this equation open up or down? What algebraic information can you use to determine this?
Ans: Opens down because the coefficient of x^2 is negative.
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2. Describe what happens to the profit as the rent per square foot is increased. Profit rises then falls as square footage increases.
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Discuss the different amounts of profit defined by the graph and the equation.
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3. Use the quadratic formula to determine the amounts that should be charged for rent that will give you zero profit. Round the value of m to the nearest tenth of a square foot.
Solve -8.3m^2 + 53.1m - 26.5 = 0
m = 0.5456 ; m = 5.852
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What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense?
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4. Will this equation give you a maximum profit or a minimum profit? How do you know?
Maximum because it has a peak.
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5. What price should be charged to reach the maximum or minimum profit?
m = -b/(2a) = -53.1/(2*-8.3) = 3.20
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6. How much profit will you have at the peak or low? Remember that the profit is measured in thousands of dollars. Include the proper units.
$58,428
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7. What is the point of the vertex? Calculate this value. How does this number relate to your answers in numbers 5 and 6?
They are the same
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8. How many solutions are there to the equation ? How do you know? Give an algebraic and a written response.
Two because the equation is a quadratic.
The discriminant is positive so there are 2 Real Number solutions.
Cheers,
Stan H.