Find the centroid of the region bounded by the given

In a truck-loading station at a post office, a small 0.200-kg package is released from rest a point A on a track that is one quarter of a circle with radius 1.60 m.The size of the packageis much less than 1.60m, so the package can be treated at aparticle. It slides down the track and reaches point B with a speed of 4.80 m/s. From point B, it slides on a level surface a distanceof 3.00 m to point C, where is comes to rest.
(a) What is the coefficient of kinetic friction on the horizontal surface?
(b) How much work is done on the package by friction as it slides down the circular arc from A to B?

A stone hangs by a fine thread from the ceiling,and a section of the same thread dangles from the bottom of the stone.If a person gives a sharp pull on the dangling thread,where is the thread likely to break:below the stone or above it?

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.

Write in words how to read each of the following out loud.
a. ${\displaystyle \{x\in R^{\prime }\mid 0<x<1\}}$
b. ${\displaystyle \{x\in R\mid x\le 0\text{or}x\Rightarrow 1\}}$
c. ${\displaystyle \{n\in Z\mid nisafactorof6\}}$
d. ${\displaystyle \{n\in Z\cdot \mid nisafactorof6\}}$

Suppose, household color TVs are replaced at an average age of ${\displaystyle \mu =7.4years}$ after purchase, and the (95% of data) range was from 5.4 to 9.4 years. Thus, the range was ${\displaystyle 9.4-5.4=4.0}$ years. Let x be the age (in years) at which a color TV is replaced. Assume that x has a distribution that is approximately normal.
(a) The empirical rule indicates that for a symmetric and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from ${\displaystyle \mu -2\sigma \to \mu +2\sigma }$ is often used for "commonly occurring" data values. Note that the interval from ${\displaystyle \mu -2\sigma \to \mu +2\sigma is4\sigma }$ in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values.
Estimating the standard deviation
For a symmetric, bell-shaped distribution,
${\displaystyle \text{standard deviation}\approx range4\approx highvalue4-lowvalue}$
where it is estimated that about 95% of the commonly occurring data values fall into this range.
Use this "rule of thumb" to approximate the standard deviation of x values, where x is the age (in years) at which a color TV is replaced. (Round your answer to one decimal place.)
(b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.)
(c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.)

Find the margin of error for the given values of ${\displaystyle c,\sigma ,}$ and n. ${\displaystyle c=0.90,\sigma =2.9,n=81}$

A person stands on a scale inside an elevator at rest. The Scale reads 800N.
a) What is the persons