Review Questions¶

Creating Expressions Equations¶

Q10.01 Create the symbolic math variables \(a\), \(b\), \(c\) and \(x\). Use these variables to define the symbolic math expressions:

\[ ax^2 + bx + c \]

\[ sin(ax) + cos(bx) + tan(cx) \]

Q10.02 Create the symbolic math variables \(a\), \(b\), \(c\) and \(x\). Use these variables to define the symbolic math equations:

\[ ax^2 + bx = c \]

\[ \frac{sin(ax)}{cos(bx)} = tan(cx) \]

Q10.03 Create the symbolic math variables \(a\), \(b\), \(c\), \(x\), and \(y\). Use these variables to define the symbolic math expression:

\[ ax^2 + bx + c \]

Substitute the variable \(y\) in for the variable \(c\).

Substitute the value 5 in for the variable \(y\).

Q10.04 Create the symbolic math variables \(E\), \(A\), \(d\), \(P\), \(L\), and \(F\). Use these variables to define the symbolic math equation:

\[ d = \frac{PL}{AE} \]

Substitute the value \(29 \times 10^6\) for \(E\)

Substitute \(F/2\) for the variable \(P\)

Q10.05 Create the symbolic math variables \(t\), \(T\), \(c\), and \(J\). Use these variables to define the symbolic math equation:

\[ t = \frac{Tc}{J} \]

Substitute the \(J = \frac{\pi}{2}c^4\) into the equation

Substitute \(T=9.0\) and \(c=4.5\). Print out the resulting value of \(t\).

Q10.06 Mohr’s circle is used in mechanical engineering to calculate the shear and normal stress. Given the height of Mohr’s circle \(\tau_{max}\) is equal to the expression below:

\[ \tau_{max} = \sqrt{(\sigma_x - \sigma_y)/2)^2 + \tau_{xy}} \]

Use SymPy expressions or equations to calculate \(\tau\) if \(\sigma_x = 90\), \(\sigma_y = 60\) and \(\tau_{xy} = 20\)

Solving Equations¶

Q10.20 Use SymPy to solve for \(x\) if \(x - 4 = 2\)

Q10.21 Use SymPy to solve for the roots of the quadratic equation \(2x^2 - 4x + 1.5 = 0\)

Q10.22 Create the symbolic math variable \(b\) and define the equation below:

\[ \frac{1}{\sqrt{2}}(b - 6) = -1 \]

Find the numeric value of \(b\) to three decimal places

Q10.30 Use SymPy to solve the system of linear equations below for the variables \(x\) and \(y\):

\[ -3x - 2y + 7 = 0 \]

\[ 5x - 3y - 6 = 0 \]

Q10.31 Use SymPy to solve the system of linear equations below for the variables \(x\), \(y\), and \(z\):

\[ 2x + 4y - z = -0.6 \]

\[ -x - 3y + 2z = 2.2 \]

\[ \frac{1}{2}x + 6y - 3z = -6.8 \]

Q10.32 A set of five equations is below:

\[ -5x_1 - 4x_2 - 2x_3 + 2x_4 + 3x_5 = 10 \]

\[ 9x_1 + 3x_2 + 4x_3 + 10x_4 + 5x_5 = -5 \]

\[ 2x_1 + 4x_2 + 3x_3 + 2x_4 + x_5 = 12 \]

\[ 5x_1 - 4x_2 + 3x_3 - 2x_4 + 2x_5 = 32 \]

\[ x_1 - x_2 + 2x_3 + 4x_4 + 3x_5 = 42 \]

Use symbolic math variables and equations to solve for \(x_1\), \(x_2\), \(x_3\), \(x_4\) and \(x_5\).

Q10.33 An equation in terms of the variables \(L\) and \(x\) is defined below.

\[ \frac{1}{6}L^3x^2 - \frac{1}{6}Lx^3 + \frac{1}{24}x^4 - \frac{1}{45}L^4 = 0 \]

Solve the equation for \(x\) in terms of the variable \(L\). Note their will be more than one solution.

Q10.50 Use SymPy to solve the system of non-linear equations below for the variables \(x\) and \(y\):

\[ 3x^2 + 2y^3 = -\frac{17}{4} \]

\[ \frac{-x^3}{2} - 8y^2 + \frac{127}{2} = 0 \]