In 1975, an airplane carrying an atomic clock flew back and forth for $15 \mathrm{~h}$ at an average speed of $140 \mathrm{~m} / \mathrm{s}$ as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. How much time did the airborne clock "lose" with respect to the clock on the ground?

Is the elastic limit a limiting force or a limiting distance?

Find the distance from the

(b). origin to the plane $2=z-x$.

A squirrel has $x$- and $y$-coordinates $(1.1 \mathrm{~m}, 3.4 \mathrm{~m})$ at time $t_1=0$ and coordinates $(5.3 \mathrm{~m},-0.5 \mathrm{~m})$ at time $t_2=3.0 \mathrm{~s}$. For this time interval, find
(b) the magnitude and direction of the average velocity.

Find the complement of the following set.

The set {x|x is an integer strictly between 0 and 8} if U is the set of all integers.

Write a program that prompts the user to enter the center of a rectangle, width, and height, and displays the rectangle.

In a pulse power circuit, the voltage of a 10$\mu$F capacitor is zero for t<0 and $v=5(1-e^{-4000t})$ V for t>0. Determine the capacitor current and the energy stored in the capacitor at t=0ms and t=10ms.

Let $\varphi$ : G $\rightarrow$ GL(V) be a representation of G. The centralizer of $\varphi$ is defined to be the set of all linear transformations, A, from V to itself such that

$$A \varphi(g)=\varphi(g) A$$

for all g $\in$ G (i.e., the linear transformations of V which commute with all $\varphi(g)^{\prime}s$). (a) Prove that a linear transformation A from V to V is in the centralizer of $\varphi$ if and only if it is an FG-module homomorphism from V to itself (so the centralizer of $\varphi$ is the same as the $\left.\operatorname{ring} \operatorname{Hom}_{F G}(V, V)\right)$. (b) Show that if z is in the center of G then $\varphi(z)$ is in the centralizer of $\varphi$. (c) Assume $\varphi$ is an irreducible representation (so V is a simple FG-module). Prove that if H is any finite abelian subgroup of GL(V) such that A$\varphi$(g) = $\varphi$(g)A for all A $\in$ H then H is cyclic (in other words, any finite abelian subgroup of the multiplicative group of units in the ring Hom$_{F G}$(V, V) is cyclic). Hom$_{F G}$(V, V) is a division ring, so this reduces to proving that a finite abelian subgroup of the multiplicative group of nonzero elements in a division ring is cyclic. Show that the division subring generated by an abelian subgroup of any division ring is a field and use Proposition 18. (d) Show that if $\varphi$ is a faithful irreducible representation then the center of G is cyclic. (e) Deduce from (d) that if G is abelian and $\varphi$ is any irreducible representation then G/ker $\varphi$ is cyclic.

Let X={a, b, c}. Define a function S from $\mathcal{P}$(X) to the set of bit strings of length 3 as follows. Let Y$\subseteq$X. If a$\in$Y, set $s_1=0$; If a$\notin$Y, set $s_1=1$; If b$\in$Y, set $s_2=0$; If b$\notin$Y, set $s_2=1$; If c$\in$Y, set $s_3=0$; If c$\in$Y, set $s_3=1$. Define S(Y)=$s_1s_2s_3$.

Prove that S is onto.

Hot exhaust gases leaving a stationary diesel engine at $450^{\circ} \mathrm{C}$ enter a $15-\mathrm{cm}$-diameter pipe at an average velocity of $7.2 \mathrm{~m} / \mathrm{s}$. The surface temperature of the pipe is $180^{\circ} \mathrm{C}$. Find the pipe length if the exhaust gases are to exit the pipe at $250^{\circ} \mathrm{C}$ after transferring heat to water in a heat recovery unit. Employ properties of air for exhaust gases.

Find the distance between each pair of points. (x+y, y) and (x-y, x)

the distance east or west from the prime meridian

The table shows the time it took several members of a cycling club to bike different distances. Sort the cyclists from least to greatest unit rate.

Find the distance between the points.

$$(-4.2,3.1),(-12.5,4.8)$$