Answer :
To solve this problem, we need to find the zeros of the function [tex]\( y = -5(x-4)^2 + 10 \)[/tex] and then interpret what these zeros represent.
### Step-by-Step Solution:
1. Understand the Function:
The given function is [tex]\( y = -5(x-4)^2 + 10 \)[/tex]. This function describes the daily profit (in hundreds of dollars) of a taco food truck, where [tex]\( x \)[/tex] is the price of a taco (in dollars).
2. Find the Zeros:
Zeros of a function are the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. So we set the function equal to zero:
[tex]\[ -5(x-4)^2 + 10 = 0 \][/tex]
3. Solve the Equation:
- First, isolate the quadratic term by subtracting 10 from both sides:
[tex]\[ -5(x-4)^2 = -10 \][/tex]
- Divide both sides by -5:
[tex]\[ (x-4)^2 = 2 \][/tex]
- Take the square root of both sides:
[tex]\[ x-4 = \pm \sqrt{2} \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 4 \pm \sqrt{2} \][/tex]
So the zeros are:
[tex]\[ x = 4 - \sqrt{2}, \quad x = 4 + \sqrt{2} \][/tex]
4. Approximate the Zeros:
- [tex]\( 4 - \sqrt{2} \approx 2.58 \)[/tex]
- [tex]\( 4 + \sqrt{2} \approx 5.41 \)[/tex]
Hence, the zeros of the function are [tex]\( x = 4 - \sqrt{2} \approx 2.58 \)[/tex] and [tex]\( x = 4 + \sqrt{2} \approx 5.41 \)[/tex].
5. Interpret the Zeros:
The zeros of the function represent the prices of tacos at which the daily profit of the food truck is [tex]$0.00. Since \( y \) represents the profit, the points where \( y = 0 \) indicate no profit. ### Answer Selection: - Zeros: A. \( x = 4 - \sqrt{2} \approx 2.58 \) and \( x = 4 + \sqrt{2} \approx 5.41 \) - Interpretation: C. The zeros are where the daily profit is $[/tex]0.00.
So, the selected answers are:
- A
- C
### Step-by-Step Solution:
1. Understand the Function:
The given function is [tex]\( y = -5(x-4)^2 + 10 \)[/tex]. This function describes the daily profit (in hundreds of dollars) of a taco food truck, where [tex]\( x \)[/tex] is the price of a taco (in dollars).
2. Find the Zeros:
Zeros of a function are the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. So we set the function equal to zero:
[tex]\[ -5(x-4)^2 + 10 = 0 \][/tex]
3. Solve the Equation:
- First, isolate the quadratic term by subtracting 10 from both sides:
[tex]\[ -5(x-4)^2 = -10 \][/tex]
- Divide both sides by -5:
[tex]\[ (x-4)^2 = 2 \][/tex]
- Take the square root of both sides:
[tex]\[ x-4 = \pm \sqrt{2} \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 4 \pm \sqrt{2} \][/tex]
So the zeros are:
[tex]\[ x = 4 - \sqrt{2}, \quad x = 4 + \sqrt{2} \][/tex]
4. Approximate the Zeros:
- [tex]\( 4 - \sqrt{2} \approx 2.58 \)[/tex]
- [tex]\( 4 + \sqrt{2} \approx 5.41 \)[/tex]
Hence, the zeros of the function are [tex]\( x = 4 - \sqrt{2} \approx 2.58 \)[/tex] and [tex]\( x = 4 + \sqrt{2} \approx 5.41 \)[/tex].
5. Interpret the Zeros:
The zeros of the function represent the prices of tacos at which the daily profit of the food truck is [tex]$0.00. Since \( y \) represents the profit, the points where \( y = 0 \) indicate no profit. ### Answer Selection: - Zeros: A. \( x = 4 - \sqrt{2} \approx 2.58 \) and \( x = 4 + \sqrt{2} \approx 5.41 \) - Interpretation: C. The zeros are where the daily profit is $[/tex]0.00.
So, the selected answers are:
- A
- C
