Question
Time (seconds) Distance (feet)
0 15.36
6 24
10 20.16
12 15.36
16 0
The table shows the distance over time that a toy drone travels when it is launched from a hilltop. Using technology, create an equation to model the situation. Make sure to provide the answer in standard form and round all values to the nearest hundredth.
(1 point)
The equation that models this situation is y=
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To model the situation described with the drone's distance over time, we can use a quadratic equation of the form:
\[ y = ax^2 + bx + c \]
where \( y \) represents the distance in feet, \( x \) represents the time in seconds, and \( a \), \( b \), and \( c \) are constants that we need to determine based on the given data points.
### Given Data Points
- (0, 15.36)
- (6, 24)
- (10, 20.16)
- (12, 15.36)
- (16, 0)
### Step 1: Setting Up Simultaneous Equations
We will substitute the \( x \) and \( y \) values from three of the data points into the quadratic equation to get three simultaneous equations.
1. Using point (0, 15.36):
\[ c = 15.36 \]
2. Using point (6, 24):
\[ 24 = 36a + 6b + 15.36 \]
\[ 36a + 6b = 24 - 15.36 \]
\[ 36a + 6b = 8.64 \]
\[ 6a + b = 1.44 \] (Equation 1)
3. Using point (10, 20.16):
\[ 20.16 = 100a + 10b + 15.36 \]
\[ 100a + 10b = 20.16 - 15.36 \]
\[ 100a + 10b = 4.8 \]
\[ 10a + b = 0.48 \] (Equation 2)
### Step 2: Solving the System of Equations
We now have two equations:
1. \( 6a + b = 1.44 \)
2. \( 10a + b = 0.48 \)
To eliminate \( b \), we can subtract the first equation from the second:
\[
(10a + b) - (6a + b) = 0.48 - 1.44
\]
\[
4a = -0.96
\]
\[
a = -0.24
\]
Now, substitute \( a \) back into Equation 1 to find \( b \):
\[
6(-0.24) + b = 1.44
\]
\[
-1.44 + b = 1.44
\]
\[
b = 2.88
\]
### Step 3: Finding \( c \)
We already know \( c = 15.36 \).
### Final Equation
Now we can write the quadratic equation:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
### Step 4: Converting to Standard Form
To write it in standard form \( Ax^2 + Bx + C \), we have:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
### Step 5: Rounding Values
All values in the equation are rounded to the nearest hundredth, resulting in the final equation:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
Thus, the final answer is:
**The equation that models this situation is:**
\[ y = -0.24x^2 + 2.88x + 15.36 \]
\[ y = ax^2 + bx + c \]
where \( y \) represents the distance in feet, \( x \) represents the time in seconds, and \( a \), \( b \), and \( c \) are constants that we need to determine based on the given data points.
### Given Data Points
- (0, 15.36)
- (6, 24)
- (10, 20.16)
- (12, 15.36)
- (16, 0)
### Step 1: Setting Up Simultaneous Equations
We will substitute the \( x \) and \( y \) values from three of the data points into the quadratic equation to get three simultaneous equations.
1. Using point (0, 15.36):
\[ c = 15.36 \]
2. Using point (6, 24):
\[ 24 = 36a + 6b + 15.36 \]
\[ 36a + 6b = 24 - 15.36 \]
\[ 36a + 6b = 8.64 \]
\[ 6a + b = 1.44 \] (Equation 1)
3. Using point (10, 20.16):
\[ 20.16 = 100a + 10b + 15.36 \]
\[ 100a + 10b = 20.16 - 15.36 \]
\[ 100a + 10b = 4.8 \]
\[ 10a + b = 0.48 \] (Equation 2)
### Step 2: Solving the System of Equations
We now have two equations:
1. \( 6a + b = 1.44 \)
2. \( 10a + b = 0.48 \)
To eliminate \( b \), we can subtract the first equation from the second:
\[
(10a + b) - (6a + b) = 0.48 - 1.44
\]
\[
4a = -0.96
\]
\[
a = -0.24
\]
Now, substitute \( a \) back into Equation 1 to find \( b \):
\[
6(-0.24) + b = 1.44
\]
\[
-1.44 + b = 1.44
\]
\[
b = 2.88
\]
### Step 3: Finding \( c \)
We already know \( c = 15.36 \).
### Final Equation
Now we can write the quadratic equation:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
### Step 4: Converting to Standard Form
To write it in standard form \( Ax^2 + Bx + C \), we have:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
### Step 5: Rounding Values
All values in the equation are rounded to the nearest hundredth, resulting in the final equation:
\[
y = -0.24x^2 + 2.88x + 15.36
\]
Thus, the final answer is:
**The equation that models this situation is:**
\[ y = -0.24x^2 + 2.88x + 15.36 \]
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.