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This SLP provides some insight into the use of quadratic formulas in business and natural science.

1.A company's costs, in millions of dollars, are given by the equation, C = x2 - 3x - 27, where x is the number of items sold, in thousands. What are the costs when 1,000 items are sold? 1,500 items?

2.A company's costs, in millions of dollars, are given by the equation, C = x2 - 3x - 27, where x is the number of items sold, in thousands. How many items must be sold for the costs to be zero?

3.An object's height, in meters, when thrown vertically from the ground at a velocity of 80 m/s is given by the equation H = -4.9t2 + 80t. Ignoring wind resistance, what is the height of the ball after 2 seconds? 3 seconds? 10 seconds?

4.An object's height, in meters, when thrown vertically from the ground at a velocity of 80 m/s is given by the equation H = -4.9t2 + 80t. Ignoring wind resistance, how many seconds will it take the object to return to the ground? In other words, when is H = 0?

5.Gatorade/Tropicana North America, a subsidiary of PepsiCo, produces fruit juices and other flavored beverages. Based on data from 1999 to 2001, the net sales (revenue) of Gatorade/Tropicana North America may be modeled by

R(t) = -107t2+496t+3452 million dollars

and the operating profit (earnings before interest and taxes) may be modeled by

P(t) = -18.5t2 + 85.5t + 433 million dollars, where t is the number of years since 1999. (Source: Modeled from 2001 PepsiCo Annual Report, p.44.)
a) Profit = Revenue - Costs. Find an expression representing the costs for this time period.
b) What was the revenue in 2000?
c) What was the profit in 2000?
d) Use the result from part a to find the costs in 2000?
e) How does this compare the result of R - P from parts b & c?

#### Solution Preview

1.
C = x^2 - 3x - 27
x = 1000 = 1 thousand
Hence, C(1) = 1^2 - 3*1 - 27 = M\$(-29 ) -- ANSWER

x= 1500 = 1.5 thousands
C(1.5) = 1.5^2 - 3*1.5 - 27 = 2.25 - 4.5 - 27 = M\$(-29.25) -- ANSWER

2. cost
C = x^2 - 3.x - 27 = 0
compare with ax^2 + bx +c = 0
a = 1, b = -3, c = -27

x = (-b + sqrt(b^2 - 4ac))/2a OR (-b - sqrt(b^2 - 4ac))/2a
=> x = ...

#### Solution Summary

Some problems related to application of quadratic equation in different fields like cost, profit and loss, and particle motion, are solved here.

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