Solve the given problems by determinants. In Exercises...

Solve the given problems by determinants. In Exercises $40-46,$ set up appropriate systems of equations. All numbers are accurate to at least two significant digits. The angle $\theta$ between two links of a robot arm is given by $\theta=a t^{3}+b t^{2}+c t,$ where $t$ is the time during an 11.8 -s cycle. If $\theta=19.0^{\circ} \mathrm{for} t=1.00 \mathrm{s}, \theta=30.9^{\circ} \mathrm{for} t=3.00 \mathrm{s},$ and $\theta=19.8^{\circ}$ for $t=5.00 \mathrm{s},$ find the equation $\theta=f(t) .$ See Fig. 5.51 (FIGURE CAN'T COPY)

Solve the given problems by determinants. In Exercises $40-46,$ set up appropriate systems of equations. All numbers are accurate to at least two significant digits.
The angle $\theta$ between two links of a robot arm is given by $\theta=a t^{3}+b t^{2}+c t,$ where $t$ is the time during an 11.8 -s cycle. If $\theta=19.0^{\circ} \mathrm{for} t=1.00 \mathrm{s}, \theta=30.9^{\circ} \mathrm{for} t=3.00 \mathrm{s},$ and $\theta=19.8^{\circ}$ for $t=5.00 \mathrm{s},$ find the equation $\theta=f(t) .$ See Fig. 5.51 (FIGURE CAN'T COPY)

Key Concepts

Polynomial Fitting

Polynomial fitting involves determining the coefficients of a polynomial function that best fits a set of data points. By plugging the data points into the polynomial form, a system of linear equations is obtained, which can then be solved using methods like determinants or Cramer’s rule to find the polynomial that exactly passes through the provided points.

Cramer’s Rule

Cramer’s rule is a mathematical theorem used to solve systems of linear equations where the number of equations is equal to the number of unknowns. It expresses each unknown as a quotient of two determinants, one being the determinant of the coefficient matrix and the other being the determinant of a matrix formed by replacing one column of the coefficient matrix with the constant values from the equations.

System of Linear Equations

This concept involves setting up a collection of equations, each representing the value of a function at specific points. In problems where the coefficients of a polynomial are unknown, known values of the function at given points allow us to form equations that can be solved simultaneously to determine these coefficients.

Determinants

Determinants are a scalar value that can be computed from the elements of a square matrix and are used to analyze certain properties of the matrix, such as invertibility. In the context of solving systems of equations, determinants help determine whether a unique solution exists and are fundamental in implementing methods like Cramer’s rule.

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