91Ó°ÊÓ

A standard number cube is tossed. Find each probability. \(P(\text { even or less than } 4)\)

Solve each equation. Check each solution. $$ \frac{1}{4 x}-\frac{3}{4}=\frac{7}{x} $$

A standard number cube is tossed. Find each probability. \(P(\text { even or prime) }\)

A standard number cube is tossed. Find each probability. \(P(\text { greater than } 1 \text { or less than } 5)\)

Write an equation for a horizontal translation of \(y=\frac{2}{x}\) Then write an equation for a vertical translation of \(y=\frac{2}{x}\) . Identify the horizontal and vertical asymptotes of the graph of each function.

Industrial Design A storage tank will have a circular base of radius \(r\) and a height of \(r .\) The tank can be either cylindrical or hemispherical (half a sphere). a. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area (including the base). For a sphere, \(V=\frac{4}{3} \pi r^{3}\) and \(S .\) A. \(=4 \pi r^{2} .\) b. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area (including the bases). c. Compare the ratios of volume to surface area for the two tanks. d. Compare the volumes of the two tanks.

Simplify. State any restrictions on the variables. $$ \frac{\left(x^{2}-x\right)^{2}}{x(x-1)^{-2}\left(x^{2}+3 x-4\right)} $$

Woodworking A tapered cylinder is made by decreasing the radius of a rod continuously as you move from one end to the other. The rate at which it tapers is the taper per foot. You can calculate the taper per foot using the formula \(T=\frac{24(R-r)}{L} .\) The lengths \(R, r,\) and \(L\) are measured in inches. a. Solve this equation for \(L\) b. Find \(L\) if \(R=4\) in. \(r=3\) in.; and \(T=0.75,0.85,\) and 0.95

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. You select a red marble and then a blue marble.

a. Critical Thinking Simplify \(\frac{\left(2 x^{n}\right)^{2}-1}{2 x^{n}-1},\) where \(x\) is an integer and \(n\) is a positive integer. \((\text { Hint: Factor the numerator.) }\) b. Use the result from part (a) to show that the value of the given expression is always an odd integer.