It is stated that a straight rod has one endpoint at the origin (0,0) and the opposite endpoint at (L,0), with a linear density defined by
, where a is a constant and x is the x coordinate.
Thus, the infinitesimal mass is expressed as:

The total mass can be calculated by integrating the above expression as follows:

Consequently, ![m=a\int\limits^L_0 {x^2} \, dx=a[\frac{x^3}{3}]_{0}^{L}=\frac{a}{3}[L^3-0]= \frac{aL^3}{3}](https://tex.z-dn.net/?f=m%3Da%5Cint%5Climits%5EL_0%20%7Bx%5E2%7D%20%5C%2C%20dx%3Da%5B%5Cfrac%7Bx%5E3%7D%7B3%7D%5D_%7B0%7D%5E%7BL%7D%3D%5Cfrac%7Ba%7D%7B3%7D%5BL%5E3-0%5D%3D%20%5Cfrac%7BaL%5E3%7D%7B3%7D)
Now, we can calculate the center of mass,
of the rod as:


Now, it follows that
x_{cm}=\frac{1}{\frac{aL^3}{3}}\int_{0}^{L}ax^3dx=\frac{3}{aL^3}\times [\frac{ax^4}{4}]_{0}^{L}
Therefore, the center of mass,
is located at:
![\frac{3}{aL^3}\times [\frac{ax^4}{4}]_{0}^{L}=\frac{3}{aL^3}\times \frac{aL^4}{4}=\frac{3}{4}L](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7BaL%5E3%7D%5Ctimes%20%5B%5Cfrac%7Bax%5E4%7D%7B4%7D%5D_%7B0%7D%5E%7BL%7D%3D%5Cfrac%7B3%7D%7BaL%5E3%7D%5Ctimes%20%5Cfrac%7BaL%5E4%7D%7B4%7D%3D%5Cfrac%7B3%7D%7B4%7DL)
Answer:
3) One-sided, compare two populations
Step-by-step explanation:
A manufacturer intends to evaluate if there is an improvement in the average gas mileage among mid-sized sedans that use a new type of tire, in comparison to those that do not utilize this tire type.
Thus, we have two distinct populations:
Population 1: Mid-sized sedans equipped with the new tire.
Population 2: Mid-sized sedans without the new tire.
Consequently, a comparison is being made between these two populations.
Furthermore, the manufacturer aims to determine if there is an enhancement in average gas mileage. As such, the alternative hypothesis will be right-tailed, indicating it is a one-sided test.
Answer:

Upon solving for y, we find:


By substitution, we have:

Thus, we arrive at:

And for x, we determine:

This gives us a length of 76 and a width of 12.5.
Step-by-step explanation:
We are dealing with a rectangle. The formula for the perimeter is:

Here, x denotes the length and y the width. The following conditions can be established:

Substituting these values yields:

When we solve for y, we find:


By substitution, we obtain:

Thus, we conclude:

The length is found to be 76, while the width measures 12.5.
Answer:
I've got no idea; I just forgot the answer, but I did my best.
Step-by-step explanation: