# algebra

David makes and sells chairs. The function p(x)=-10x^2 + 100x - 210 indicates how much profit he makes in a month if he sells the chairs for 10 - x dollars each.

What should David charge per chair to make the maximum profit, and what is the maximum profit he can make in a month?

a) \$40 at \$5 per chair**
b) \$50 at \$7 per chair
c) \$40 at \$7 per chair
d) \$50 at \$5 per chair
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I need help on this, I do not know where to start. After looking on a graphing calculator, I think the answer may be a....

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1. you want the max value of -10x^2 + 100x - 210
= -10(x^2 - 10x + ... - .....) - 210
I am completing the square
= -10(x^2 - 10x + 25-25) - 210
= -10( (x-5)^2 - 25) - 210
= -10(x-5)^2 + 250 - 210
= -10(x-5)^2 + 40

Max is 40 when x = 5

how does that fit into the given answers?
remember you want to calculate 10-x for the selling price of a chair.

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2. a x² + bx + c

The graph of a quadratic function is a parabola.

The lowest or the highest point on a parabola is called the vertex.

For:

a > 0

parabola is opens up and have minimum value in vertex.

For:

a < 0

parabola is opens down and have maximum value in vertex.

The vertex has the x - coordinate:

x = - b / 2 a

In this case:

p(x) = - 10 x² + 100 x - 210

a = - 10 , b = 100 , c = - 210

Vertex x - coordinate:

x = - b / 2 a = - 100 / 2 ∙ ( - 10 ) = - 100 / - 20 = 5

p(x) = - 10 x² + 100 x - 210

For x = 5

p(5) = - 10 ∙ 5² + 100 ∙ 5 - 210 =

- 10 ∙ 25 + 500 - 210 =

- 250 + 500 - 210 = 40

Maximum profit is at the vertex of the parabola :

x = 5 , p = 40

The price is 10 - x = \$10 - \$ 5 = \$5

so:

\$40 at \$5 per chair

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3. Thank you so much, guys!! You have really made me understand this now. I appreciate it!
Have a great night! :)

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2. 👎

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