We appreciate your visit to AP Precalculus Chapter 1 Test A 2 A physics student obtains the following data involving a ball rolling down an inclined plane where tex t. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
Sure! Let's walk through how to find an algebraic model that fits the given data for the elapsed time [tex]\( t \)[/tex] and the distance traveled [tex]\( y \)[/tex].
### Step-by-Step Solution:
1. Identify the Given Data:
- We have the following pairs of data points representing time [tex]\( t \)[/tex] and distance [tex]\( y \)[/tex]:
[tex]\( ( t = 1, y = 2 ) \)[/tex]
[tex]\( ( t = 2, y = 6 ) \)[/tex]
[tex]\( ( t = 3, y = 12 ) \)[/tex]
[tex]\( ( t = 4, y = 20 ) \)[/tex]
[tex]\( ( t = 5, y = 30 ) \)[/tex]
2. Determine the Nature of the Data:
- Typically, the distance [tex]\( y \)[/tex] traveled by a ball rolling down an inclined plane as a function of time [tex]\( t \)[/tex] can be modeled using a quadratic function. A quadratic function has the general form:
[tex]\[
y = at^2 + bt + c
\][/tex]
- We need to find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] that best fit our data.
3. Fit a Quadratic Model:
- Using methods such as polynomial regression, we fit a quadratic model to the given data points. The result of this fitting process gives us the coefficients for the quadratic equation.
4. Results:
- From the fitting process, we obtain the following coefficients:
[tex]\[
a = 1.0, \quad b = 1.0, \quad \text{and} \quad c \approx 2.66127661 \times 10^{-14}
\][/tex]
- The coefficient [tex]\( c \)[/tex] is very close to zero, so it does not significantly affect the equation.
5. Form the Algebraic Model:
- Plugging these coefficients into the quadratic equation, we get:
[tex]\[
y = 1.0 t^2 + 1.0 t + 2.66127661 \times 10^{-14}
\][/tex]
6. Simplify the Model:
- Since [tex]\( 2.66127661 \times 10^{-14} \)[/tex] is very small, we can approximate it as zero for simplicity. Therefore, our algebraic model becomes:
[tex]\[
y = t^2 + t
\][/tex]
### Conclusion:
The algebraic model that fits the given data for the distance traveled by a ball rolling down an inclined plane as a function of elapsed time is:
[tex]\[
y = t^2 + t
\][/tex]
### Step-by-Step Solution:
1. Identify the Given Data:
- We have the following pairs of data points representing time [tex]\( t \)[/tex] and distance [tex]\( y \)[/tex]:
[tex]\( ( t = 1, y = 2 ) \)[/tex]
[tex]\( ( t = 2, y = 6 ) \)[/tex]
[tex]\( ( t = 3, y = 12 ) \)[/tex]
[tex]\( ( t = 4, y = 20 ) \)[/tex]
[tex]\( ( t = 5, y = 30 ) \)[/tex]
2. Determine the Nature of the Data:
- Typically, the distance [tex]\( y \)[/tex] traveled by a ball rolling down an inclined plane as a function of time [tex]\( t \)[/tex] can be modeled using a quadratic function. A quadratic function has the general form:
[tex]\[
y = at^2 + bt + c
\][/tex]
- We need to find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] that best fit our data.
3. Fit a Quadratic Model:
- Using methods such as polynomial regression, we fit a quadratic model to the given data points. The result of this fitting process gives us the coefficients for the quadratic equation.
4. Results:
- From the fitting process, we obtain the following coefficients:
[tex]\[
a = 1.0, \quad b = 1.0, \quad \text{and} \quad c \approx 2.66127661 \times 10^{-14}
\][/tex]
- The coefficient [tex]\( c \)[/tex] is very close to zero, so it does not significantly affect the equation.
5. Form the Algebraic Model:
- Plugging these coefficients into the quadratic equation, we get:
[tex]\[
y = 1.0 t^2 + 1.0 t + 2.66127661 \times 10^{-14}
\][/tex]
6. Simplify the Model:
- Since [tex]\( 2.66127661 \times 10^{-14} \)[/tex] is very small, we can approximate it as zero for simplicity. Therefore, our algebraic model becomes:
[tex]\[
y = t^2 + t
\][/tex]
### Conclusion:
The algebraic model that fits the given data for the distance traveled by a ball rolling down an inclined plane as a function of elapsed time is:
[tex]\[
y = t^2 + t
\][/tex]
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