# Algebra

david makes and sells chairs. The function p(x)=-10x^2+100x-210, graphed below, indicated 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?

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1. the maximum profit is at the vertex of the parabola. As you know, the vertex of the parabola

ax^2+bx+c

is at

x = -b/2a
y = (b^2-4ac)/4a

So, for

-10x^2+100x-210

the vertex is at (5,40)

Since x=5, the price is 10-x = 5 as well.

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2. p(x) is a parabola opening down(sheds water)
therefore the vertex is at the top p
So use complete square to find vertex (if you do not know any calculus)

-10 x^2 + 100 x = p + 210

x^2 - 10 x = - (1/10)(p+210)
x^2 - 10 x + 25= -(1/10)(p+210)+25

(x-5)^2 = -(1/10)(p+210-250)
x = 5 and p = 40
10-5 = 5
max p = 40

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3. The answer is A. \$40 at \$5 per chair.

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