To find the selling price(s) of the T-shirt that would generate a daily profit of \[tex]$50, we need to set the given quadratic function equal to 50 and solve for \( x \).
The quadratic function given is:
\[ y = -10x^2 + 160x - 430 \]
1. Formulating the equation:
We need to find the \( x \)-values where the daily profit \( y \) is \$[/tex]50. So, we set [tex]\( y \)[/tex] to 50:
[tex]\[ 50 = -10x^2 + 160x - 430 \][/tex]
2. Rearranging the equation:
To make it easier to solve, we rearrange the equation to the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ 50 = -10x^2 + 160x - 430 \][/tex]
Subtract 50 from both sides to set the equation to zero:
[tex]\[ -10x^2 + 160x - 430 - 50 = 0 \][/tex]
Simplify:
[tex]\[ -10x^2 + 160x - 480 = 0 \][/tex]
Now the equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where:
[tex]\[ a = -10, \quad b = 160, \quad c = -480 \][/tex]
3. Choosing the method to solve:
The equation [tex]\( -10x^2 + 160x - 480 = 0 \)[/tex] is a quadratic equation. We can solve a quadratic equation using several methods, such as factoring, completing the square, or using the quadratic formula. Here, we use the quadratic formula because it provides a straightforward way to find the roots for any quadratic equation.
4. Quadratic formula:
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
5. Discriminant:
First, we calculate the discriminant [tex]\( \Delta \)[/tex] (inside the square root part of the quadratic formula):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Using the values [tex]\( a = -10, b = 160, c = -480 \)[/tex]:
[tex]\[ \Delta = 160^2 - 4 \cdot (-10) \cdot (-480) \][/tex]
[tex]\[ \Delta = 25600 - 19200 \][/tex]
[tex]\[ \Delta = 6400 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] equals 6400.
6. Finding the solutions:
Using the quadratic formula with the discriminant:
[tex]\[ x = \frac{-160 \pm \sqrt{6400}}{2 \cdot -10} \][/tex]
Calculate the square root of the discriminant:
[tex]\[ \sqrt{6400} = 80 \][/tex]
Now, the two potential solutions for [tex]\( x \)[/tex] are:
[tex]\[ x_1 = \frac{-160 + 80}{-20} = \frac{-80}{-20} = 4 \][/tex]
[tex]\[ x_2 = \frac{-160 - 80}{-20} = \frac{-240}{-20} = 12 \][/tex]
7. Conclusion:
The solutions to the equation [tex]\( -10x^2 + 160x - 480 = 0 \)[/tex] are [tex]\( x = 4 \)[/tex] and [tex]\( x = 12 \)[/tex]. Therefore, the selling prices for the T-shirt that would generate a daily profit of \[tex]$50 are \( \$[/tex]4 \) and [tex]\( \$12 \)[/tex].
Thus, the selling prices that would generate a \$50 daily profit are [tex]\( 4 \)[/tex] dollars and [tex]\( 12 \)[/tex] dollars.
