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Home Question Bank Other Archive 2024 December 08

Other Questions from Dec 08,2024

Browse the Other Q&A Archive for Dec 08,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Solve the following linear programming problem. Restrict \( x \geq 0 \) and \( y \geq 0 \) \[ \begin{array}{r}\text { Minimize } g=50 x+80 y \text { subject to the following. } \\ \qquad \begin{array}{l} 11 x+15 y \geq 255 \\ x+3 y \geq 33\end{array} \\ (x, y)=\square\end{array} \] 5 Explain how the relationship between the Sun, Earth, and the Moon creates the Moon's phases. 3 If a crescent Moon is less than half, what is a gibbous Moon? (A) Half (B) More than half Find the eigenvalues of the matrix \( A=\left[\begin{array}{ccc}9 & -6 & -5 \\ 0 & -2 & 8 \\ 0 & 0 & 6\end{array}\right] \). Orthogonally diagonalize the matrix, giving an orthogonal matrix \( P \) and a diagonal matrix \( D \). To save time, the eigenvalues are -7 and 11 . \[ A=\left[\begin{array}{rrr}1 & 4 & -8 \\ 4 & -5 & -4 \\ -8 & -4 & 1\end{array}\right] \] Enter the matrices P and D below. Evaluate the determinant of the matrix \( A=\left[\begin{array}{ccc}8 & 0 & 7 \\ -11 & 7 & 0 \\ 9 & -7 & 7\end{array}\right] \). Minimize the required number of computations by carefully choosing a row or column to expand along, and use the properties of determinants to simplify the process. Find the relative size \( T \) of the shock wave of earthquakes with magnitude 5.4 , as measured on the Richter scale (Answer in scientific notation, rounded to the nearest hundredth). Let \( \mathbf{y}=\left[\begin{array}{l}2 \\ 7 \\ 1\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{c}\frac{2}{3} \\ \frac{1}{3} \\ \frac{2}{3}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}-\frac{2}{3} \\ \frac{2}{3} \\ \frac{1}{3}\end{array}\right] \), and \( \mathrm{W}= \) Span \( \left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\} \). Complete parts (a) and (b). b. Compute proj \( \mathbf{W}_{\mathbf{W}} \mathbf{y} \) and \( \left(\mathrm{UU}^{\top}\right) \mathbf{y} \). Let \( \mathbf{y}=\left[\begin{array}{l}2 \\ 7 \\ 1\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{c}\frac{2}{3} \\ \frac{1}{3} \\ \frac{2}{3}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}-\frac{2}{3} \\ \frac{2}{3} \\ \frac{1}{3}\end{array}\right] \), and \( \mathrm{W}= \) Span \( \left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\} \). Complete parts (a) and (b). a. Let \( \mathrm{U}=\left[\begin{array}{ll}\mathbf{u}_{1} & \mathbf{u}_{2}\end{array}\right] \). Compute \( \mathrm{U}^{\top} \mathrm{U} \) and \( \mathrm{U} \mathrm{U}^{\mathrm{T}} \). 3. Determine the null space of the following matrix: \[ \left(\begin{array}{cccc}-1 & -3 & 1 & 0 \\ 3 & 0 & 6 & 9\end{array}\right) \]
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