- Determine the required profit from football sales: \(400 - 232.50 = 167.50\).
- Set up the equation \(-4x^2 + 80x - 150 = 167.50\) and simplify to \(4x^2 - 80x + 317.50 = 0\).
- Apply the quadratic formula to find the two prices that satisfy the condition.
- The two prices are approximately \(\$14.54\) and \(\$5.46\), which correspond to the given options.
- The final answers are \(\boxed{\$14.54, \$5.46}\).
Explanation
- Determine the Required Profit and Set Up the Equation
To determine the price of footballs, we first calculate the necessary profit the store needs to earn from football sales alone. The store already earns \(\$232.50\) from selling soccer balls. Since the manager's total daily profit goal is \(\$400.00\), the required profit from football sales (\(P_f\)) is given by:
\[P_f = 400.00 - 232.50 = 167.50.\]
The daily profit from selling footballs at price \(x\) is modeled by the function \(y = -4x^2 + 80x - 150\). We set this model equal to the required profit:
\[-4x^2 + 80x - 150 = 167.50.\]
- Solve the Quadratic Equation
To solve for \(x\), we first rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\):
\[-4x^2 + 80x - 150 - 167.50 = 0\]
\[-4x^2 + 80x - 317.50 = 0.\]
Multiplying the entire equation by \(-1\) gives:
\[4x^2 - 80x + 317.50 = 0.\]
Now, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 4\), \(b = -80\), and \(c = 317.50\):
\[x = \frac{-(-80) \pm \sqrt{(-80)^2 - 4(4)(317.50)}}{2(4)}.\]
Calculating the discriminant:
\[(-80)^2 = 6400\]
\[4 \times 4 \times 317.50 = 5080\]
\[6400 - 5080 = 1320.\]
Thus, the roots are:
\[x = \frac{80 \pm \sqrt{1320}}{8} = \frac{80 \pm 36.3318}{8}.\]
Solving for the two values:
\[x_1 = \frac{80 + 36.3318}{8} \approx 14.54\]
\[x_2 = \frac{80 - 36.3318}{8} \approx 5.46.\]
Therefore, the two prices are approximately \(\$14.54\) and \(\$5.46\).
Examples
Understanding quadratic profit models is a common task for business managers and entrepreneurs. By knowing the relationship between price and profit, they can identify optimal price points to reach specific revenue goals. For example, if a clothing store models their profit per item based on sales trends, they can solve similar equations to determine the best price for their new seasonal collection to maximize or hit target profits. This mathematical approach helps in making data-driven decisions for pricing strategies.