# SOLUTION: The members of the Lincoln High School Prom Committee are trying to raise money for their senior prom. They plan to sell teddy bears. The senior advisor told them that the profit e

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 Question 253584: The members of the Lincoln High School Prom Committee are trying to raise money for their senior prom. They plan to sell teddy bears. The senior advisor told them that the profit equation for their project is y= -0.1x^2+9x-50 where x is the price at which the teddy bears will be sold and y is the profit, in dollars. How much profit can the committee expect to make if they sell the teddy bears for $20 each? What price should they charge for the teddy bears to make maximum profit possible? It is a word problem with 2 questions! I did the first one: y= -0.1(20)^2+9(20)-50 =-40+180-50 =90 If the teddy bears were sold for 20 dollars each, they would make 90 dollars! I don't understand the second problem: What price should they charge for the teddy bears to make maximum profit possible? Found 3 solutions by JimboP1977, jim_thompson5910, solver91311:Answer by JimboP1977(308) (Show Source): You can put this solution on YOUR website!They are asking you at what value x will give the largest value of y. Quadratics always produce a graph called a parabola. In this case the parabola is inverted (upside down) because the coefficient of x^2 is a minus number. In essence, we need to find when the parabola peaks or when the gradient is zero. We can do this by differentiation. dy/dx = -0.2x+9.This gives the gradient at any given value of x. We want the gradient to be zero so 0= -0.2x+9 so x = 45. If we plot the graph of y= -0.1x^2+9x-50 we can see that this is true. Answer by jim_thompson5910(28476) (Show Source): You can put this solution on YOUR website!The max profit occurs at the vertex since the vertex represents the max value of y. In order to find the vertex, we first need to find the x-coordinate of the vertex. To find the x-coordinate of the vertex, use this formula: . Start with the given formula. From , we can see that , , and . Plug in and . Multiply 2 and to get . Divide. So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also . Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex. Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So the y-coordinate of the vertex is . So the vertex is . Since the vertex represents the max value of y, and y is the profit, this means that the max profit is$152.50 (since the y value of the vertex is y=152.5) This profit is attained when the price is \$45 (since the x coordinate of the vertex is x=45) Answer by solver91311(16868)   (Show Source): You can put this solution on YOUR website! Since you have a 2nd degree function with a negative lead coefficient, the graph will be a parabola opening down. The horizontal coordinate of the vertex of the parabola is the input value that gives the maximum function value. The horizontal coordinate of a parabola is found by dividing the additive inverse of the 1st degree term coefficient by two times the lead coefficient, which is to say , or in your case . You can do your own arithmetic. John