Math / Data & Statistics

QuestionThe table below shows the energy of a 10 kg ball as a function of velocity. How much energy will the ball have at $10 \mathrm{~m} / \mathrm{s}$ ? (Type the number, do not include units in your answer)
Use a matrix to run a quadratic regression \begin{tabular}{|c|c|} \hline Velocity (m/s) & Energy (Joules) \\ \hline 0 & 0 \\ \hline 2 & 12 \\ \hline 3 & 27 \\ \hline 6 & 108 \\ \hline 10 & $?$ \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The relationship between velocity and energy is quadratic.
2. We will use the given data points to perform a quadratic regression.
3. The quadratic equation will be in the form $E = av^2 + bv + c$ .

STEP 2

1. Set up the system of equations using the given data points.
2. Solve the system using matrix methods to find the coefficients $a$ , $b$ , and $c$ .
3. Use the quadratic equation to calculate the energy at $10 \, \text{m/s}$ .

STEP 3

Write the quadratic equation for each data point:
For $v = 0$ , $E = 0$ : $0 = a(0)^2 + b(0) + c$ $\Rightarrow c = 0$
For $v = 2$ , $E = 12$ : $12 = a(2)^2 + b(2) + c$ $12 = 4a + 2b + c$
For $v = 3$ , $E = 27$ : $27 = a(3)^2 + b(3) + c$ $27 = 9a + 3b + c$
For $v = 6$ , $E = 108$ : $108 = a(6)^2 + b(6) + c$ $108 = 36a + 6b + c$

STEP 4

Substitute $c = 0$ into the equations:
$12 = 4a + 2b$ $27 = 9a + 3b$ $108 = 36a + 6b$

STEP 5

Convert the equations to matrix form:
$\begin{bmatrix} 4 & 2 \\ 9 & 3 \\ 36 & 6 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 12 \\ 27 \\ 108 \end{bmatrix}$

STEP 6

Use matrix methods (e.g., Gaussian elimination or inverse matrix) to solve for $a$ and $b$ .
After solving, we find: $a = 1$ $b = 0$

STEP 7

Substitute $a = 1$ , $b = 0$ , and $c = 0$ into the quadratic equation:
$E = 1 \cdot v^2 + 0 \cdot v + 0$ $E = v^2$
Calculate the energy at $v = 10 \, \text{m/s}$ :
$E = 10^2 = 100$
The energy at $10 \, \text{m/s}$ is:
$\boxed{100}$

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Studdy Solution

STEP 1

1. The relationship between velocity and energy is quadratic.
2. We will use the given data points to perform a quadratic regression.
3. The quadratic equation will be in the form $E = av^2 + bv + c$ .

STEP 2

1. Set up the system of equations using the given data points.
2. Solve the system using matrix methods to find the coefficients $a$ , $b$ , and $c$ .
3. Use the quadratic equation to calculate the energy at $10 \, \text{m/s}$ .

STEP 3

STEP 4

Substitute $c = 0$ into the equations:
$12 = 4a + 2b$ $27 = 9a + 3b$ $108 = 36a + 6b$

STEP 5

Convert the equations to matrix form:
$\begin{bmatrix} 4 & 2 \\ 9 & 3 \\ 36 & 6 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 12 \\ 27 \\ 108 \end{bmatrix}$

STEP 6

Use matrix methods (e.g., Gaussian elimination or inverse matrix) to solve for $a$ and $b$ .
After solving, we find: $a = 1$ $b = 0$

STEP 7

Substitute $a = 1$ , $b = 0$ , and $c = 0$ into the quadratic equation:
$E = 1 \cdot v^2 + 0 \cdot v + 0$ $E = v^2$
Calculate the energy at $v = 10 \, \text{m/s}$ :
$E = 10^2 = 100$
The energy at $10 \, \text{m/s}$ is:
$\boxed{100}$

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Studdy Solution

STEP 1

1. The relationship between velocity and energy is quadratic.2. We will use the given data points to perform a quadratic regression.3. The quadratic equation will be in the form E=av2+bv+c E = av^2 + bv + c E=av2+bv+c.

STEP 2

1. Set up the system of equations using the given data points.2. Solve the system using matrix methods to find the coefficients a a a, b b b, and c c c.3. Use the quadratic equation to calculate the energy at 10 m/s 10 \, \text{m/s} 10m/s.

STEP 3

STEP 4

Substitute c=0 c = 0 c=0 into the equations:12=4a+2b 12 = 4a + 2b 12=4a+2b 27=9a+3b 27 = 9a + 3b 27=9a+3b 108=36a+6b 108 = 36a + 6b 108=36a+6b

STEP 5

Convert the equations to matrix form:[4293366][ab]=[1227108]\begin{bmatrix} 4 & 2 \\ 9 & 3 \\ 36 & 6 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 12 \\ 27 \\ 108 \end{bmatrix}⎣⎡​4936​236​⎦⎤​[ab​]=⎣⎡​1227108​⎦⎤​

STEP 6

Use matrix methods (e.g., Gaussian elimination or inverse matrix) to solve for a a a and b b b.After solving, we find: a=1 a = 1 a=1 b=0 b = 0 b=0

STEP 7

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1. The relationship between velocity and energy is quadratic.
2. We will use the given data points to perform a quadratic regression.
3. The quadratic equation will be in the form $E = av^2 + bv + c$ .

1. Set up the system of equations using the given data points.
2. Solve the system using matrix methods to find the coefficients $a$ , $b$ , and $c$ .
3. Use the quadratic equation to calculate the energy at $10 \, \text{m/s}$ .

Substitute $c = 0$ into the equations:
$12 = 4a + 2b$ $27 = 9a + 3b$ $108 = 36a + 6b$

Convert the equations to matrix form:
$\begin{bmatrix} 4 & 2 \\ 9 & 3 \\ 36 & 6 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 12 \\ 27 \\ 108 \end{bmatrix}$

Use matrix methods (e.g., Gaussian elimination or inverse matrix) to solve for $a$ and $b$ .
After solving, we find: $a = 1$ $b = 0$