Random variables X and Y have joint PDFf_{X,Y}(x,y)=\begin{cases}12e^{-(

The magnetic field ${\displaystyle \overrightarrow{B}}$ in acertain region is 0.128 ,and its direction is that of the z-axis in the figure.

Part A
What is the magnetic flux across the surface abcd in the figure?
Part B
What is the magnetic flux across the surface befc ?
Part C What is the magnetic flux across the surface aefd?
Part D
What is the net flux through all five surfaces that enclose the shaded volume?

A 30.0 kg block is resting on a flat horizontal table. On topof this block is resting a 15.0 kg block, to which a horizontal spring is attached. The spring constant of the spring is 325 N/m. The coefficient of kinetic friction between the lower block and the table is 0.600, and the coefficient of static friction between the two blocks is 0.900. A horizontal force is applied to the lower block. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block,determine (a) the amount by which the spring is compressed and (b) the magnitude of the force F.

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig. 1) The rope will break if the tension in it exceeds $4.50\cdot {10}^4$N and our hero's mass is 86.5 kg.

 Figure 1 a) If the angle $\theta$ is ${13.0}^{\circ }$, find the tension iin the rope.

b) What is the smallest value the angle $\theta$ can have if the rope is not to break?

A 95.0kg person stands on a scale in an elevator. What is the apparent weight when the elevator is (a) accelerating upward with an acceleration of 1.80 ${\displaystyle \frac{m}{s^2}}$, (b) moving upward at a constant speed and (c) accelerating downward with an acceleration of 1.30 ${\displaystyle \frac{m}{s^2}}$?

A wagon with two boxes of Gold, having total mass 300 kg, is cutloose from the hoses by an outlaw when the wagon is at rest 50m upa 6.0 degree slope. The outlaw plans to have the wagon roll downthe slope and across the level ground, and then fall into thecanyon where his confederates wait. But in a tree 40m from thecanyon edge wait the Lone Ranger (mass 75.0kg) and Tonto (mass60.0kg). They drop vertically into the wagon as it passes beneaththem. a) if they require 5.0 s to grab the gold and jump out, willthey make it before the wagon goes over the edge? b) When the twoheroes drop into the wagon, is the kinetic energy of the system ofthe heroes plus the wagon conserved? If not, does it increase ordecrease and by how much?

The joint density of X and Y is given by,
${\displaystyle f_{XY}(x,y)=\frac{6}{7}(x^2+\frac{xy}{2})0<x<1,0<y<2}$
a) Calculate ${\displaystyle P(X<1,Y>1)}$
b) Find the marginal probability distributions of X and Y.
c) Find the conditional probability density function of Y given ${\displaystyle X=0.5}$ and calculate $P(Y<1/X=0.5)$

Graph the quadratic function ${\displaystyle f\left(x\right)={(x+1)}^2-2}$. Give the (a) vertex, (b) axis, (c) domain, and (d) range.
Then determine (e) the largest open interval of the domain over which the function is increasing and (f) the largest open interval over which the function is decreasing.
(a) The vertex is (-1,-2).
(Type an ordered pair.)
Use the graphing tool to graph the function.
(b) The axis is
(Type an equation.)