Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
