The article “Mechanistic-Empirical Design of Bituminous Roads: An Indian Perspective” presents an equation of the form $y=a\left(1 / x_{1}\right)^{b}\left(1 / x_{2}\right)^{c}$ for predicting the number of repetitions for laboratory fatigue failure (y) in terms of the tensile strain at the bottom of the bituminous beam $(x_1)$ and the resilient modulus $(x_2).$ Transform this equation into a linear model, and express the linear model coefficients in terms of a, b, and c.

It is difficult to graph a linear function by hand. One method that works if the $x$, $y$, and $z$-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure $12.68$; this shows the part of the plane in the octant where $x \geq 0, y \geq 0, z \geq 0.$ If the intercepts are not all positive, the same method works if the $x$, $y$, and $z$-axes are drawn from a different perspective. Use this method to graph the linear function.

$$z= 2 -2x + y$$

The machine part shown in Figure P5.73 is 10-mm thick, is made of AISI 1020 cold-rolled steel (see Appendix D for properties), and is subjected to a tensile load of $P=45 \mathrm{kN}$. Determine the minimum radius $r$ that can be used between the two sections if a factor of safety of 2 with respect to failure by yield is specified. Round the minimum fillet radius up to the nearest 1-mm multiple.

Figure shows a steel bar fastened to a rigid ground plane with two $0.25-\text{in}$-dia hardened steel dowel pins. For $P=1500\ \mathrm{lb}$, find using FEA:

(a) The shear stress in each pin. (b) The direct bearing stress in each pin and hole. (c) The minimum value of dimension $h$ to prevent tearout failure if the steel bar has a shear strength of $32.5\ \mathrm{kpsi}$.

An experiment to investigate the survival time in hours of an electronic component consists of placing the parts in a test cell and running them for 100 hours under elevated temperature conditions. (This is called an “accelerated” life test.) Eight components were tested with the following resulting failure times:

$$75 63 100^+ 36 51 45 80 90$$

The observation

$$100^+$$

indicates that the unit still functioned at 100 hours. Is there any meaningful measure of location that can be calculated for these data? What is its numerical value?

When complete P elements are injected into embryos from an M strain, they transpose into the chromosomes of the germ line, and progeny reared from these embryos can be used to establish new P strains. However, when complete P elements are injected into embryos from insects that lack these elements, such as mosquitoes, they do not transpose into the chromosomes of the germ line. What does this failure to insert in the chromosomes of other insects indicate about the nature of P element transposition?

Solve the wave equation on the real line for the given initial position f and initial velocity g, first by separation of variables and the Fourier integral, and then by using the Fourier transform. The same solution should result from both methods. $c=1, g(x)=0,$ and

$$f(x)=\left\{\begin{array}{ll} 2-|x| & \text { for }-2 \leq x \leq 2 \\ 0 & \text { for }|x|>2 \end{array}\right.$$

Given that $F(\omega)=\mathcal{F}[f(t)]$, prove the following results, using the definition of Fourier transform: (a) $\mathcal{F}\left[f\left(t-t_{0}\right)\right]=e^{-j \omega t_{0}} F(\omega)$ (b) $\mathcal{F}\left[\frac{d f(t)}{d t}\right]=j \omega F(\omega)$ (c) $\mathcal{F}[f(-t)]=F(-\omega)$ (d) $\mathcal{F}[t f(t)]=j \frac{d}{d \omega} F(\omega)$

Why is using your sociological imagination an example of using a sociological perspective?

Stress Test A machine part was tested by bending it $x$ centimeters ten times per minute until the time $y$ (in hours) of failure. The results are recorded in the table.

Use the regression capabilities of a graphing utility to find a linear model for the data.

Draw a cube in one-point perspective so that the vanishing point is to the right of the cube.

It is difficult to graph a linear function by hand. One method that works if the $x$, $y$, and $z$-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure $12.68$; this shows the part of the plane in the octant where $x \geq 0, y \geq 0, z \geq 0.$ If the intercepts are not all positive, the same method works if the $x$, $y$, and $z$-axes are drawn from a different perspective. Use this method to graph the linear function.

$$z= 4+x-2y$$

(a) Find the Laplace transforms of (i) $\cos (\omega t+\phi) \quad \text { (ii) } \mathrm{e}^{-\omega t} \sin (\omega t+\phi)$ (b) Using Laplace transform methods, solve the differential equation $\frac{d^{2} x}{d t^{2}}+4 \frac{d x}{d t}+8 x=\cos 2 t$ given that x = 2 and dx/dt = 1 when t = 0.

The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase a six-year-old automobile, with a working voltage regulator and plan to own it for six years. a. What is the probability that the voltage regulator fails during your ownership? b. If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure?