Construct the general solution of x'=Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

$$A=\left[ \begin{array}{rrr}{53} & {-30} & {-2} \\ {90} & {-52} & {-3} \\ {20} & {-10} & {2}\end{array}\right]$$

Kears's sonnet is Petrarchan, a form consisting of an octave (eight lines) and a sestet (six lines). In his poem, how is the content of the octave and the sestet related?

Reread the footnote for aboriginal. What does the footnote explain that helps you understand the people's response to the tsunami?

Solve each inequality and graph the solutions. $\frac{1}{2}|x+5| \geq 2$

Use substitution to find the Taylor series at x = 0 of the function.

$$\frac { 1 } { 2 - x }$$

A particle moving in a planar force field has a position vector x that satisfies x'=Ax. The $2 \times 2$ matrix A has eigenvalues 4 and 2, with corresponding eigenvectors

$$\mathbf{v}_{1}=\left[ \begin{array}{r}{-3} \\ {1}\end{array}\right]$$

and

$$\mathbf{v}_{2}=\left[ \begin{array}{r}{-1} \\ {1}\end{array}\right]$$

. Find the position of the particle at time t, assuming that

$$\mathbf{x}(0)=\left[ \begin{array}{r}{-6} \\ {1}\end{array}\right]$$

.

Approximate the positive value of the constant a such that the area enclosed by y=cosh ax, y=0, x=0, and x=1 is 2 square units. Express your answer to at least five decimal places.

Dalton's atomic theory states that (1) all atoms of an element are positively charged, (2) different elements can have the same mass, (3) atoms of a given element must be identical, (4) all the atoms in a compound are identical.

Student Council has a budget of $1000 for the homecoming dance. So far, they have spent$350 dollars for music. a. Write an equation to represent the amount of money left to spend. Then solve the equation. b. They then spent $225 on decorations. Write an equation to represent the amount of money left. c. If the Student Council spent their entire budget, write an equation to represent how many$6 tickets they must sell to make a profit.

Suppose that the mass of a spherical mothball decreases with time, due to evaporation, at a rate that is proportional to its surface area. Assuming that it always retains the shape of a sphere, it can be shown that the radius $r$ of the sphere decreases linearly with the time t.
(a) If, at a certain instant, the radius is 0.80$\mathrm { mm }$ and 4 days later it is 0.75$\mathrm { mm }$ , find an equation for $r$(in millimeters) in terms of the elapsed time $t$(in days). (b) How long will it take for the mothball to completely evaporate?

Simplify the expression. Write your answer using exponents.

$$( 17 \cdot 16 ) ^ { 4 }$$

Ben earns $8.82 for the first 40 h he works each week and 1.5 times his hourly pay for any hours over 40. Write a formula that can calculate the total weekly pay for Ben when he works more than 40 h.

Use your calculator to find the total weekly pay for Ben if he works the following number of hours. 42

Simplify (2 + x)(y + 5).