Compare the fractions in each pair. Think about how close the fractions are to $0, \frac{1}{2}, \text { or } 1$. a. $\frac{5}{8} \text { and } \frac{7}{12}$ b. $\frac{14}{15} \text { and } \frac{7}{8}$ c. $\frac{1}{9} \text { and } \frac{1}{23}$

Rewrite $\frac{2(x+5)-4}{2}-x$ using the fewest terms possible. Justify each step with a property or operation.

A spacecraft is approaching a space station that is orbiting Earth. When the craft is 1000 km from the space station, reverse thrusters must be applied to begin braking. The time, t, in hours, required to reach a distance, d, in kilometres, from the space station while the thrusters are being fired can be modelled by $t=\log _{0.5}\left(\frac{d}{1000}\right)$. The docking sequence can be initialized once the craft is within 10 km of the station’s docking bay. a) How long after the reverse thrusters are first fired should docking procedures begin? b) What are the domain and range of this function? What do these features represent?

Determine the product of all values of k for which the polynomial equation $2 x^{3}-9 x^{2}+12 x-k=0$ has a double root.

One cycle of a sine function begins at $x=-\frac{\pi}{4}$ and ends at $x=\frac{\pi}{3}$. a) Determine the period of the function. b) Determine the phase shift of the function.

From the data shown, determine the economic service lives of the defender and challenger. (Note: Values in the table are AW values, not individual year-end values.)

$$\begin{array}{ccc} \text { Years Retained } & \text { AW of Defender, } \$ & \text { AW of Challenger, } \$ \\ \hline 1 & -145,000 & -136,000 \\ 2 & -96,429 & -126,000 \\ 3 & -63,317 & -92,000 \\ 4 & -39,321 & -53,000 \\ 5 & -49,570 & -38,000 \\ \hline \end{array}$$

Describe the similarities and differences between the parabola $y=x^{2}$ and power functions with even degree greater than two. Use graphs to support your answer.

Which of the following sequences are the first four terms of an arithmetic sequence? For those that are, write down the value of the common difference. (a) 7 10 13 16 ... (b) 3 5 9 15 ... (c) 1 0.1 0.01 0.001 .. . (d) 4 2 0 -2 ... (e) 2 -3 4 -5 ... (f) p - 2q p - q p p + q ... (g) $\frac{1}{2} a \quad \frac{1}{3} a \quad \frac{1}{4} a \quad \frac{1}{5} a \quad \ldots$ (h) x 2x 3x 4x ...

The Gap has some of its jeans stone-washed under a contract with Vietnam Garment Corporation (VGC). If VGC’s estimated operating cost per machine is $26,000 for year 1 and it increases by$1500 per year through year 5, the equivalent uniform annual cost per machine over years 1 to 5, at an interest rate of 8% per year, is closest to: (a) $30,850 (b)$28,770 (c) $26,930 (d)$23,670

Give the maximum number of turning points of the graph of each function. $f(x)=x^{5}-9 x^{2}$

Two large flat metal sheets are located at z = 0 and z = d and are maintained at 0 and $V_{0}$, respectively. The charge density between the sheets is $\rho_{v}(z)=\rho_{o} z / d$, where $\boldsymbol{\rho}_{\boldsymbol{o}}$ is a constant. Determine the potential at all points between the plates.

The campus recruiter for an international conglomerate classifies the large number of students she interviews into three categories-the lower quarter, the middle half, and the upper quarter. If she meets six students on a given morning, what is the probability that they will be evenly divided among the three categories? What is the marginal probability that exactly two will belong to the middle half?

The problem is based on the following cash flows and an interest rate of 10% per year.

$$\begin{array}{lcc} \text { Alternative } & \mathrm{X} & \mathrm{Y} \\ \hline \text { First cost, } \$ & -200,000 & -800,000 \\ \text { Annual cost, } \$ \text {/year } & -60,000 & -10,000 \\ \text { Salvage value, } \$ & 20,000 & 150,000 \\ \text { Life, years } & 5 & \infty \\ \hline \end{array}$$

The annual worth of perpetual service for alternative X is represented by the following equation: (a) −200,000(0.10) − 60,000 + 20,000(0.10) (b) −200,000(A/P,10%,5) − 60,000 + 20,000(A/F,10%,5) (c) −200,000(A/P,10%,5) − 60,000 − 20,000(A/F,10%,5) (d) −200,000(0.10) − 60,000 + 20,000(A/F,10%,5)

Decide whether each statement is true or false. If false, tell why. The graph of a nonzero function cannot be symmetric with respect to the x-axis.