In previous question, let the height h equal $6.00$ m and the speed v$_{boy}$ equal $2.00$ m/s. Assume that the food pack starts from rest.

(a) Tabulate and graph the speed–time graph.

(b) Tabulate and graph the acceleration–time graph. Let the range of time be from $0$ s to $5.00$ s and the time intervals be $0.500$ s.

An RC series circuit consists of an AC source, a $33.0 - \mu \text F$ capacitor, and a 30.0-$\Omega$ resistor. If the source emf is given by $\varepsilon = \varepsilon_ { max } \sin ( \omega t )$, where $E _ { \max } = 180.0 \mathrm { V }$ and $\omega = 20.0 \mathrm { s } ^ { - 1 }$, what is the current amplitude?

Find the distance between the two points. If necessary, you may draw graphs but you shouldn't need to use the distance formula.

$(0,0)$ and $(3,4)$

Find the distance between the points.

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

If H is a subgroup of G and A is an abelian group let $M_{G / H}(A)$ denote the abelian group of all maps from the left cosets gH of H in G to A. (a) Prove that

$$M_{1}^{G}(A) \cong M_{1}^{H}\left(M_{G / H}(A)\right)$$

as H-modules. [If $\left\{g_{i}\right\}_{i \in \mathcal{I}}$ is a choice of left coset representatives of H in G define the correspondence between $f \in M_{1}^{G}(A)$ and

$$F: H \rightarrow M_{G / H}(A)$$

by

$$F(h)\left(g_{i} H\right)=f\left(g_{i} h\right),$$

and check that this is an isomorphism of H-modules. (b) A G-module A such that $H^{n}(H, A)=0$ for all n $\geq$ 1 and all subgroups H of G is called cohomologically trivial. Prove that $M_{1}^{G}(A)$ is a cohomologically trivial for any abelian group A. (c) If G is finite, prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A$ is cohomologically trivial for all abelian groups A.

the distance east or west from the prime meridian

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?

Riley and Rhoda plan to buy several bags of dog food and a dog collar. Each bags of dog food costs $7, and the dog collar costs$5. a. Use the information above to write an expression for the amount of money they will spend if they buy b bags of food. b. Riley and Rhoda have $30. Use this information and the expression you wrote for Part (a) to write an inequality for finding the number of bags of food they could buy. c. Solve the inequality and graph the solutions. Check your answer in the original situation.

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.

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

Write the correct word in the space next to each definition.

random $\rule{2cm}{0.15mm}$

Find the least squares regression line for the points $\{(1, 1), (2, 0), (5, −5) \}$. Graph the points and the line.

Set U: integers from 1 to 20, Set P: prime numbers, Set T: multiples of 3, Set F: multiples of 5. Create a Venn diagram of these sets.

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.