Question
Use the table to answer the question.
Time (seconds) and Distance (feet) of a Toy Drone's Flight
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=
.
Time (seconds) and Distance (feet) of a Toy Drone's Flight
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
GPT-4o mini
To model the distance of the toy drone over time using the given data points, we can create a quadratic equation since the pattern suggests a parabolic trajectory due to the nature of the flight.
The data points from the table are:
- (0, 15.36)
- (6, 24)
- (10, 20.16)
- (12, 15.36)
- (16, 0)
We will use the form of a quadratic equation:
\[ y = ax^2 + bx + c \]
Where:
- \( y \) represents the distance in feet,
- \( x \) represents the time in seconds.
To find the coefficients \( a \), \( b \), and \( c \), we can use three points from the data set. Let's use the points (0, 15.36), (6, 24), and (10, 20.16) for our computations.
1. **Using the first point (0, 15.36):**
\[ 15.36 = a(0)^2 + b(0) + c \]
\[ c = 15.36 \]
2. **Using the second point (6, 24):**
\[ 24 = a(6)^2 + b(6) + 15.36 \]
\[ 24 - 15.36 = 36a + 6b \]
\[ 8.64 = 36a + 6b \]
\[ 36a + 6b = 8.64 \quad \text{(1)} \]
3. **Using the third point (10, 20.16):**
\[ 20.16 = a(10)^2 + b(10) + 15.36 \]
\[ 20.16 - 15.36 = 100a + 10b \]
\[ 4.80 = 100a + 10b \]
\[ 100a + 10b = 4.80 \quad \text{(2)} \]
Now we have two equations (1) and (2):
1. \( 36a + 6b = 8.64 \)
2. \( 100a + 10b = 4.80 \)
We can solve these equations simultaneously.
Let's simplify equation (1) by dividing all terms by 6:
\[ 6a + b = 1.44 \quad \text{(3)} \]
Now let's express \( b \) in terms of \( a \):
\[ b = 1.44 - 6a \]
Substituting this into equation (2):
\[ 100a + 10(1.44 - 6a) = 4.80 \]
\[ 100a + 14.4 - 60a = 4.80 \]
\[ 40a = 4.80 - 14.4 \]
\[ 40a = -9.6 \]
\[ a = -0.24 \]
Now substitute \( a \) back into equation (3) to find \( b \):
\[ 6(-0.24) + b = 1.44 \]
\[ -1.44 + b = 1.44 \]
\[ b = 2.88 \]
Now we have:
- \( a = -0.24 \)
- \( b = 2.88 \)
- \( c = 15.36 \)
Thus, the equation is:
\[ y = -0.24x^2 + 2.88x + 15.36 \]
To express this in standard form, we can represent it as:
\[ y = -0.24x^2 + 2.88x + 15.36 \]
Rounding to the nearest hundredth, the final equation is:
\[ y = -0.24x^2 + 2.88x + 15.36 \]
The data points from the table are:
- (0, 15.36)
- (6, 24)
- (10, 20.16)
- (12, 15.36)
- (16, 0)
We will use the form of a quadratic equation:
\[ y = ax^2 + bx + c \]
Where:
- \( y \) represents the distance in feet,
- \( x \) represents the time in seconds.
To find the coefficients \( a \), \( b \), and \( c \), we can use three points from the data set. Let's use the points (0, 15.36), (6, 24), and (10, 20.16) for our computations.
1. **Using the first point (0, 15.36):**
\[ 15.36 = a(0)^2 + b(0) + c \]
\[ c = 15.36 \]
2. **Using the second point (6, 24):**
\[ 24 = a(6)^2 + b(6) + 15.36 \]
\[ 24 - 15.36 = 36a + 6b \]
\[ 8.64 = 36a + 6b \]
\[ 36a + 6b = 8.64 \quad \text{(1)} \]
3. **Using the third point (10, 20.16):**
\[ 20.16 = a(10)^2 + b(10) + 15.36 \]
\[ 20.16 - 15.36 = 100a + 10b \]
\[ 4.80 = 100a + 10b \]
\[ 100a + 10b = 4.80 \quad \text{(2)} \]
Now we have two equations (1) and (2):
1. \( 36a + 6b = 8.64 \)
2. \( 100a + 10b = 4.80 \)
We can solve these equations simultaneously.
Let's simplify equation (1) by dividing all terms by 6:
\[ 6a + b = 1.44 \quad \text{(3)} \]
Now let's express \( b \) in terms of \( a \):
\[ b = 1.44 - 6a \]
Substituting this into equation (2):
\[ 100a + 10(1.44 - 6a) = 4.80 \]
\[ 100a + 14.4 - 60a = 4.80 \]
\[ 40a = 4.80 - 14.4 \]
\[ 40a = -9.6 \]
\[ a = -0.24 \]
Now substitute \( a \) back into equation (3) to find \( b \):
\[ 6(-0.24) + b = 1.44 \]
\[ -1.44 + b = 1.44 \]
\[ b = 2.88 \]
Now we have:
- \( a = -0.24 \)
- \( b = 2.88 \)
- \( c = 15.36 \)
Thus, the equation is:
\[ y = -0.24x^2 + 2.88x + 15.36 \]
To express this in standard form, we can represent it as:
\[ y = -0.24x^2 + 2.88x + 15.36 \]
Rounding to the nearest hundredth, the final equation is:
\[ y = -0.24x^2 + 2.88x + 15.36 \]