Write an original problem that can be solved using the Fundamental Counting Principle. Then solve the problem.

Let a, b, r be constants. Show that

$$y = C e ^ { - k t } + a + b k \left( \frac { k \sin rt - r \cos r t } { k ^ { 2 } + r ^ { 2 } } \right)$$

is a general solution of

$$\frac { d y } { d t } = - k ( y - a - b \sin r t )$$

Write an original problem that can be solved using the Fundamental Counting Principle. Then solve the problem.

A car model comes in nine colors, with or without air conditioning, with or without a sun roof, with or without automatic transmission, and with or without antilock brakes. In how many ways can the car be ordered with regard to these options?

Write the solutions of the inequality in set-builder notation. 12 - 8m ≥ 60

Find the effective rate corresponding to $10\frac{5}{8}\% / \text{year}$ compounded monthly.

A connecting rod is supported by a knife-edge at point A. For small oscillations the angular acceleration of the connecting rod is governed by the relation $\alpha=-6 \theta$, where $\alpha$ is expressed in$rad/s^2$ and $\theta$ in radians. Knowing that the connecting rod is released from rest when $\theta = 20^o$, determine (a) the maximum angular velocity, (b) the angular position when t = 2 s.

Consider 1.0 L of a solution that is $0.85\ \mathrm { M }\ \mathrm { HOC } _ { 6 } \mathrm { H } _ { 5 }$ and $0.80\ \mathrm {M}\ \mathrm { NaOC } _ { 6 } \mathrm { H } _ { 5 }.$ $( K _ { \mathrm { a } }$ for $\mathrm { HOC } _ { 6 } \mathrm { H } _ { 5 } = 1.6 \times 10 ^ { - 10 }. )$ a. Calculate the pH of this solution. b. Calculate the pH after 0.10 mole of HCl has been added to the original solution. Assume no volume change on addition of HCl. c. Calculate the pH after 0.20 mole of NaOH has been added to the original buffer solution. Assume no volume change on addition of NaOH.

Compare acellular slime molds, cellular slime molds, and water molds.

Make a box-and-whisker plot of the data. Find the inter-quartile range. {12, 15, 12, 6, 18, 29}.

Find each sum or difference. Simplify if possible.

$$1 \frac { 3 } { 5 } + 2 \frac { 7 } { 8 }$$

What is the transfer function of a filter? Describe how the transfer function of a filter can be determined using laboratory methods.

Solve by using the strategy Consider Extreme Cases. One side of a parallelogram has length 4 cm, and an adjacent side has length 6 cm. Is it possible for the area of this parallelogram to be less than 0.2 $\mathrm{cm}^{2}$.

Evaluate the following integrals, which have irreducible quadratic factors. $\int \frac{x}{(x-1)\left(x^{2}+2 x+2\right)^{2}} d x$