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....
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.
a x² + bx + c
is called a quadratic function.
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
Answer a) is correct.
Thank you so much, guys!! You have really made me understand this now. I appreciate it!
Have a great night! :)
To find the price per chair that will maximize David's profit, we need to find the vertex of the quadratic function p(x) = -10x^2 + 100x - 210. The x-coordinate of the vertex represents the price per chair that will maximize the profit.
To find the vertex, we can either use the formula x = -b/2a or complete the square. Let's use the formula x = -b/2a since it's simpler in this case.
From the given function p(x) = -10x^2 + 100x - 210, we can see that a = -10 and b = 100. Plugging these values into the vertex formula:
x = -100 / (2 * -10)
x = -100 / -20
x = 5
So, the price per chair that will maximize the profit is $5.
To find the maximum profit, we substitute the value of x = 5 back into the function p(x):
p(5) = -10(5)^2 + 100(5) - 210
p(5) = -250 + 500 - 210
p(5) = 40
Hence, the maximum profit David can make in a month is $40.
Therefore, the correct answer is a) $40 at $5 per chair.