Let's call :
x=the number of downloaded standard version
y= the number of downloaded high-quality version
We have : -the high-quality version was downloaded three times as often as the
standard version => 3x=y
-standard version is 2.8 megabytes .the high-quality version is 4.6
MB.The total size downloaded for the two versions was 4482 MB
=> 2.8x+4.6y=4482
Solve for x,y
x=270 , y=810
Answer:
![x^2y^{11} / 16](https://tex.z-dn.net/?f=x%5E2y%5E%7B11%7D%20%2F%2016)
Step-by-step explanation:
Hope this helped!
Answer: D) all American school teachers.
Step-by-step explanation:
In statistics, A population is a very large set of individuals or values that belong to a specified group established by researcher.
A sample is a subset of population that represents the entire population in research.
Given : We want to know the opinions of American school teachers about establishing a national test for high school graduation.
Clearly , by the above definition
Population = all American school teachers.
You obtain a list of the members of the National Education Association and mail a questionnaire to 2500 teachers chosen at random from this list.
sample = 2500 teachers chosen at random from this list.
Hence, the intended population is all American school teachers.
Thus , the correct answer is D) all American school teachers.
Answer:a range of values so defined that there is a specified probability that the value of a parameter lies within it.
hope this helps
Answer:
The absolute maximum is
and the absolute minimum value is ![0.](https://tex.z-dn.net/?f=0.)
Step-by-step explanation:
Differentiate of
both sides w.r.t.
,
![f(t)=2 \cos t+\sin 2t](https://tex.z-dn.net/?f=f%28t%29%3D2%20%5Ccos%20t%2B%5Csin%202t)
![\Rightarrow f'(t)=-2\sin t+2\cos 2t](https://tex.z-dn.net/?f=%5CRightarrow%20f%27%28t%29%3D-2%5Csin%20t%2B2%5Ccos%202t)
Now take ![f'(t)=0](https://tex.z-dn.net/?f=f%27%28t%29%3D0)
![\Rightarrow -2\sin t+2\cos 2t=0](https://tex.z-dn.net/?f=%5CRightarrow%20-2%5Csin%20t%2B2%5Ccos%202t%3D0)
![\Rightarrow 2\cos 2t=2\sin t](https://tex.z-dn.net/?f=%5CRightarrow%202%5Ccos%202t%3D2%5Csin%20t)
![\Rightarrow \cos 2t=\sin t](https://tex.z-dn.net/?f=%5CRightarrow%20%5Ccos%202t%3D%5Csin%20t)
![\Rightarrow 1-2\sin ^2t =\sin t \quad \quad [\because \cos 2t = 1-2\sin ^2t]](https://tex.z-dn.net/?f=%5CRightarrow%201-2%5Csin%20%5E2t%20%3D%5Csin%20t%20%20%5Cquad%20%5Cquad%20%20%5B%5Cbecause%20%5Ccos%202t%20%3D%201-2%5Csin%20%5E2t%5D)
![\Rightarrow 2\sin ^2t+\sin t-1=0](https://tex.z-dn.net/?f=%5CRightarrow%202%5Csin%20%5E2t%2B%5Csin%20t-1%3D0)
![\Rightarrow 2\sin ^2t+2\sin t-\sin t-1=0](https://tex.z-dn.net/?f=%5CRightarrow%202%5Csin%20%5E2t%2B2%5Csin%20t-%5Csin%20t-1%3D0)
![\Rightarrow 2\sin t(\sin t+1)-1(\sin t+1)=0](https://tex.z-dn.net/?f=%5CRightarrow%202%5Csin%20t%28%5Csin%20t%2B1%29-1%28%5Csin%20t%2B1%29%3D0)
![\Rightarrow (\sin t+1)(2\sin t-1)=0](https://tex.z-dn.net/?f=%5CRightarrow%20%28%5Csin%20t%2B1%29%282%5Csin%20t-1%29%3D0)
![\Rightarrow \sin t+1=0 \;\text{and}\; 2\sin t-1=0](https://tex.z-dn.net/?f=%5CRightarrow%20%5Csin%20t%2B1%3D0%20%20%5C%3B%5Ctext%7Band%7D%5C%3B%202%5Csin%20t-1%3D0)
![\Rightarrow \sin t =-1 \;\text{and}\; \sin t =\frac 12](https://tex.z-dn.net/?f=%5CRightarrow%20%5Csin%20t%20%3D-1%20%20%5C%3B%5Ctext%7Band%7D%5C%3B%20%20%20%5Csin%20t%20%3D%5Cfrac%2012)
In the interval
, the answer to this problem is ![\frac {\pi}6](https://tex.z-dn.net/?f=%5Cfrac%20%7B%5Cpi%7D6)
Now find the second derivative of
w.r.t.
,
![f''(t)=-2\cos t-4\sin 2t](https://tex.z-dn.net/?f=f%27%27%28t%29%3D-2%5Ccos%20t-4%5Csin%202t)
![\Rightarrow \left[f''(t)\right]_{t=\frac {\pi}6}=-2\times \frac {\sqrt 3}2-4\times \frac{\sqrt 3}2=-3\sqrt 3](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cleft%5Bf%27%27%28t%29%5Cright%5D_%7Bt%3D%5Cfrac%20%7B%5Cpi%7D6%7D%3D-2%5Ctimes%20%5Cfrac%20%7B%5Csqrt%203%7D2-4%5Ctimes%20%5Cfrac%7B%5Csqrt%203%7D2%3D-3%5Csqrt%203)
Thus,
is maximum at
and minimum at ![t=0](https://tex.z-dn.net/?f=t%3D0)
![\left[f(t)\right]_{t=\frac {\pi}6}=2\times \frac {\sqrt 3}2+\frac{\sqrt 3}2=\frac{3\sqrt 3}2\;\text{and}\;\left[f(t)\right]_{t=\frac{\pi}2}= 2\times 0+0=0](https://tex.z-dn.net/?f=%5Cleft%5Bf%28t%29%5Cright%5D_%7Bt%3D%5Cfrac%20%7B%5Cpi%7D6%7D%3D2%5Ctimes%20%5Cfrac%20%7B%5Csqrt%203%7D2%2B%5Cfrac%7B%5Csqrt%203%7D2%3D%5Cfrac%7B3%5Csqrt%203%7D2%5C%3B%5Ctext%7Band%7D%5C%3B%5Cleft%5Bf%28t%29%5Cright%5D_%7Bt%3D%5Cfrac%7B%5Cpi%7D2%7D%3D%202%5Ctimes%200%2B0%3D0)
Hence, the absolute maximum is
and the absolute minimum value is
.