Question

Steven needs to buy \(t\) theme park tickets for himself and his family. Each ticket costs $$\$ 80$$, and the number of tickets he needs to buy can be modeled by the expression \(t^2-4 t-90=6\) when \(t>0\). What is the total cost of the theme park tickets that Steven purchased? A) $$\$ 640$$ B) $$\$ 800$$ C) $$\$ 960$$ D) $$\$ 1,120$$ $$ \begin{aligned} & 2 c+3 d=17 \\ & 6 c+5 d=39 \end{aligned} $$

Short answer

The total cost of the theme park tickets Steven purchased is C) \( \$960\).

Step-by-step solution

Solve the quadratic equation for positive t

First, we need to solve the given quadratic equation \(t^2 - 4t - 90 = 6\). To do this, let's first subtract 6 from both sides of the equation: \(t^2 - 4t - 96 = 0\) Now let's find the factors of the given quadratic equation or use the quadratic formula, which is given by: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -96\). Substitute the values into the formula: \(t = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-96)}}{2(1)}\)

Simplify the expression and determine the correct value of t

Now, we will simplify the expression to find the possible values of \(t\): \(t = \frac{4 \pm \sqrt{16 + 384}}{2}\) \(t = \frac{4 \pm \sqrt{400}}{2}\) \(t = \frac{4 \pm 20}{2}\) There are two possible solutions for \(t\): \(t = \frac{4 + 20}{2} = \frac{24}{2} = 12\) \(t = \frac{4 - 20}{2} = \frac{-16}{2} = -8\) Since the problem states that \(t > 0\), we will choose the positive solution, \(t=12\).

Find the total cost of the tickets

Now that we have found the number of tickets Steven needs to purchase, \(t=12\), we will multiply this by the cost of each ticket, \(80\), to find the total cost: Total cost = Number of tickets * Cost of each ticket Total cost = \(12 * \$80 = \$960\)

Match the answer with the given options

Compare our calculated total cost of \$960 with the given options and choose the matching one: A) \( \$640\) B) \( \$800\) C) \( \$960\) D) \( \$1,120\) The correct answer is C) \( \$960\).