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Sagot :
Sure! Let's address each part of the question step by step.
### 1. Find the zeros of the function.
The function given is [tex]\( y = (-16 t - 2)(t - 1) \)[/tex]. To find the zeros of this function, we need to determine for which values of [tex]\( t \)[/tex] the function [tex]\( y \)[/tex] equals zero.
Set [tex]\( y \)[/tex] to zero and solve for [tex]\( t \)[/tex]:
[tex]\[ (-16 t - 2)(t - 1) = 0 \][/tex]
This equation is satisfied if either of the factors is zero.
[tex]\[ -16 t - 2 = 0 \quad \text{or} \quad t - 1 = 0 \][/tex]
Solve each equation separately:
1. [tex]\( -16 t - 2 = 0 \)[/tex]
[tex]\[ -16 t = 2 \][/tex]
[tex]\[ t = -\frac{1}{8} \][/tex]
2. [tex]\( t - 1 = 0 \)[/tex]
[tex]\[ t = 1 \][/tex]
Therefore, the zeros of the function are [tex]\( t = -\frac{1}{8} \)[/tex] and [tex]\( t = 1 \)[/tex].
### 2. What do the zeros tell us in this situation? Are both zeros meaningful?
In this context, the zeros of the function represent the times at which the height [tex]\( y \)[/tex] of the beach ball is zero. This means the beach ball reaches the ground at these times.
Let's interpret the zeros:
- [tex]\( t = -\frac{1}{8} \)[/tex]: This is a negative time, which typically does not have physical meaning because time cannot be negative in this real-world scenario. It suggests the ball was at ground level before it was even thrown, which is not meaningful.
- [tex]\( t = 1 \)[/tex]: This is a positive time, indicating that the ball reaches the ground 1 second after being thrown. This is a meaningful result as it represents the time it takes for the ball to hit the ground after it was released.
### 3. From what height is the beach ball thrown?
To find the height from which the beach ball is thrown, we evaluate the function [tex]\( y \)[/tex] at [tex]\( t = 0 \)[/tex].
Substitute [tex]\( t = 0 \)[/tex] into the equation [tex]\( y = (-16 t - 2)(t - 1) \)[/tex]:
[tex]\[ y(0) = (-16 \cdot 0 - 2)(0 - 1) \][/tex]
[tex]\[ y(0) = (-2)(-1) \][/tex]
[tex]\[ y(0) = 2 \][/tex]
Therefore, the beach ball is thrown from a height of 2 units (the specific unit, such as meters or feet, is not specified in the problem).
### Summary
- The zeros of the function are [tex]\( t = -\frac{1}{8} \)[/tex] and [tex]\( t = 1 \)[/tex].
- [tex]\( t = -\frac{1}{8} \)[/tex] is not meaningful in this context, but [tex]\( t = 1 \)[/tex] indicates the ball hits the ground after 1 second.
- The beach ball is thrown from a height of 2 units.
### 1. Find the zeros of the function.
The function given is [tex]\( y = (-16 t - 2)(t - 1) \)[/tex]. To find the zeros of this function, we need to determine for which values of [tex]\( t \)[/tex] the function [tex]\( y \)[/tex] equals zero.
Set [tex]\( y \)[/tex] to zero and solve for [tex]\( t \)[/tex]:
[tex]\[ (-16 t - 2)(t - 1) = 0 \][/tex]
This equation is satisfied if either of the factors is zero.
[tex]\[ -16 t - 2 = 0 \quad \text{or} \quad t - 1 = 0 \][/tex]
Solve each equation separately:
1. [tex]\( -16 t - 2 = 0 \)[/tex]
[tex]\[ -16 t = 2 \][/tex]
[tex]\[ t = -\frac{1}{8} \][/tex]
2. [tex]\( t - 1 = 0 \)[/tex]
[tex]\[ t = 1 \][/tex]
Therefore, the zeros of the function are [tex]\( t = -\frac{1}{8} \)[/tex] and [tex]\( t = 1 \)[/tex].
### 2. What do the zeros tell us in this situation? Are both zeros meaningful?
In this context, the zeros of the function represent the times at which the height [tex]\( y \)[/tex] of the beach ball is zero. This means the beach ball reaches the ground at these times.
Let's interpret the zeros:
- [tex]\( t = -\frac{1}{8} \)[/tex]: This is a negative time, which typically does not have physical meaning because time cannot be negative in this real-world scenario. It suggests the ball was at ground level before it was even thrown, which is not meaningful.
- [tex]\( t = 1 \)[/tex]: This is a positive time, indicating that the ball reaches the ground 1 second after being thrown. This is a meaningful result as it represents the time it takes for the ball to hit the ground after it was released.
### 3. From what height is the beach ball thrown?
To find the height from which the beach ball is thrown, we evaluate the function [tex]\( y \)[/tex] at [tex]\( t = 0 \)[/tex].
Substitute [tex]\( t = 0 \)[/tex] into the equation [tex]\( y = (-16 t - 2)(t - 1) \)[/tex]:
[tex]\[ y(0) = (-16 \cdot 0 - 2)(0 - 1) \][/tex]
[tex]\[ y(0) = (-2)(-1) \][/tex]
[tex]\[ y(0) = 2 \][/tex]
Therefore, the beach ball is thrown from a height of 2 units (the specific unit, such as meters or feet, is not specified in the problem).
### Summary
- The zeros of the function are [tex]\( t = -\frac{1}{8} \)[/tex] and [tex]\( t = 1 \)[/tex].
- [tex]\( t = -\frac{1}{8} \)[/tex] is not meaningful in this context, but [tex]\( t = 1 \)[/tex] indicates the ball hits the ground after 1 second.
- The beach ball is thrown from a height of 2 units.
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