To determine the zeros of the function [tex]\( f(x) = -x^2 - 7x + 30 \)[/tex] and what they represent, let's proceed with a detailed, step-by-step solution.
1. Identify the function:
The function given is [tex]\( f(x) = -x^2 - 7x + 30 \)[/tex].
2. Set the function equal to zero to find the zeros:
We need to solve for [tex]\( x \)[/tex] such that:
[tex]\[
-x^2 - 7x + 30 = 0.
\][/tex]
3. Solve the quadratic equation:
Solving [tex]\( -x^2 - 7x + 30 = 0 \)[/tex], we get the following solutions:
[tex]\[
x = -10 \quad \text{and} \quad x = 3.
\][/tex]
4. Interpret the zeros:
The zeros of the function [tex]\( x = -10 \)[/tex] and [tex]\( x = 3 \)[/tex] are the values of [tex]\( x \)[/tex] where the vertical distance [tex]\( f(x) \)[/tex] is zero. This means the diver hits the water at these points.
5. Select the meaningful zero:
In the context of the physical scenario, only the positive zero makes sense because negative distance ( [tex]\( x = -10 \)[/tex] ) does not apply here. Therefore, the relevant zero is [tex]\( x = 3 \)[/tex].
6. Conclusion:
When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 0 \)[/tex]. This indicates that the diver hits the water 3 feet away horizontally from the diving board.
So, the correct interpretation is:
[tex]\[
x = 3; \text{ the diver hits the water 3 feet away horizontally from the board.}
\][/tex]
The choices are as follows:
- [tex]\( x = 10 \)[/tex]; the diver hits the water 10 feet away horizontally from the board.
- [tex]\( x = 3 \)[/tex]; the diver hits the water 3 feet away horizontally from the board.
- [tex]\( x = 10 \)[/tex]; the diver jumps in the pool at 10 feet per second.
- [tex]\( x = 3 \)[/tex]; the diver jumps in the pool at 3 feet per second.
Given the context and the physical meaning, the correct answer is:
[tex]\[
x = 3; \text{ the diver hits the water 3 feet away horizontally from the board.}
\][/tex]
