Question

the price is right
the state fair is in town! the fair is selling ticket packages for the rides it offers. the package includes passes for the ferris wheel, roller coaster, bumper cars, and merry - go - round. the ticket sales can be modeled by the function ( p(x)=-25x^{2}+525x ), where ( x ) is the price of the ticket. without creating graphs, answer the following.

1. rewrite ( p(x) ) in its equivalent vertex form. what is the vertex of this function and what does in mean in relation to ticket sales?
2. using vertex form, write an equation that can be used to determine the ticket price that gives a revenue of $0.
3. solve the equation you wrote in #2. show or explain your work.
4. using vertex form, write an equation that can be used to determine the approximate ticket price that gives revenue of $1200.
5. solve the equation you write in #4. show or explain your work.
6. katherine and garret wanted to know the prices of a ticket package that would generate a revenue of $1450. to do this, katherine wrote the equation: ( -25x^{2}+525x = 1450 ) and said, \i can complete the square to solve this equation by factoring out a - 25 first.\ garret wrote the equation ( -25x^{2}+525x + 1450 = 0 ) and said, \i can use factoring and the zero - product property to solve this problem.\ who do you agree with? explain or justify your answer then solve the equation.
7. what price should the carnival sell the ticket packages for? why?

Explanation:

Step1: Analyze Katherine's and Garret's equations

Katherine's equation: \(-25x^{2}+525x = 1450\). Garret's equation: \(-25x^{2}+525x + 1450=0\). The standard form of a quadratic equation is \(ax^{2}+bx + c = 0\). Katherine's equation is not in standard form. To use factoring or the zero - product property, the equation should be in standard form.

Step2: Rewrite Katherine's equation

Starting with \(-25x^{2}+525x=1450\), subtract \(1450\) from both sides to get \(-25x^{2}+525x - 1450 = 0\). Divide through by \(-25\) (as Katherine suggested) gives \(x^{2}-21x + 58=0\).

Step3: Solve the quadratic equation \(x^{2}-21x + 58 = 0\)

Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the equation \(ax^{2}+bx + c = 0\). Here \(a = 1\), \(b=-21\), \(c = 58\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(-21)^{2}-4\times1\times58=441 - 232=209\). Then \(x=\frac{21\pm\sqrt{209}}{2}\approx\frac{21\pm14.46}{2}\). So \(x_1=\frac{21 + 14.46}{2}=\frac{35.46}{2}=17.73\) and \(x_2=\frac{21-14.46}{2}=\frac{6.54}{2}=3.27\)

Answer:

Agree with Katherine. The solutions of the equation \(-25x^{2}+525x = 1450\) (after rewriting and using the quadratic formula) are \(x\approx3.27\) and \(x\approx17.73\)