Question

Help!!! A business can manufacture 50 unicycles a week and sell all of them for $200 each. The owner is considering increasing the price of the unicycles, but she knows that it will decrease sales. She uses this equation to estimate how much in dollars, y, she will make if she raises the price by x dollars:

y=10,000+50x−x^2
If she wants to make $10,6000, what is the least amount she can raise the price of each unicycle?

Answer 1

Answer:  20 dollars

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Explanation:

I'm assuming you meant to write $10,600 instead of $10,6000.

We'll replace y with this value and solve for x.

$y=10,000+50x-x^2\\\\10,600=10,000+50x-x^2\\\\0=10,000+50x-x^2-10,600\\\\-x^2+50x-600 = 0$

Use the quadratic formula with a = -1, b = 50, c = -600

$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-50\pm\sqrt{(50)^2-4(-1)(-600)}}{2(-1)}\\\\x = \frac{-50\pm\sqrt{100}}{-2}\\\\x = \frac{-50\pm10}{-2}\\\\x = \frac{-50+10}{-2} \ \text{ or } \ x = \frac{-50-10}{-2}\\\\x = \frac{-40}{-2} \ \text{ or } \ x = \frac{-60}{-2}\\\\x = 20 \ \text{ or } \ x = 30$

We go for the smaller x value because the question asks "what is the least amount she can raise the price?".

If she raises the price by $20, then she would estimate to make about $10,600.