A company sells shoes for $40 per pair if less than 50 pair are ordered. If more than 50 are ordered (up to 600), the price is reduced at a rate of $.04 times the number oredered. How many pairs can a dealer purchase for $8400
Solution
Let x = number of shoes which can be purchased by $8,400
Therefore,
($40-$0.04x)x = 8,400
40x - 0.04x^2 = 8400
The above equation is a simple quadratic equation.
Divide both sides of the equation by 0.04 to get
1,000x - x^2 = 210,000
Multiply both sides by -1 to get
x^2 - 1000x = -210000
Transform the above quadratic equation to get
x^2 - 1000x + 210000 = 0
Solve for the roots of the quadratic equation by thinking of two numbers when multiplied equals 210,000 and when added equals 1000 --- 700 and 300!
Hence, (x - 700)(x - 300) = 0
Therefore,
x1 = 700
x2 = 300
If you test the above numbers by substituting them in the quadratic equation 40x - 0.04x^2 = 8400, both numbers arrive at $8,400. This means that if you buy 300 pairs of shoes, you will pay $8,400 AND if you buy 700 pairs of shoes, you will still pay $8,400!
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