<u>Given</u>:
The two functions are
, ![r(x)=x^2+1](https://tex.z-dn.net/?f=r%28x%29%3Dx%5E2%2B1)
We need to determine the value of
and ![(q \circ r)(-3)](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28-3%29)
<u>Value of </u>
<u>:</u>
Let us determine the value of ![(r \circ q)(-3)](https://tex.z-dn.net/?f=%28r%20%5Ccirc%20q%29%28-3%29)
![(r \circ q)(x)=r[q(x)]](https://tex.z-dn.net/?f=%28r%20%5Ccirc%20q%29%28x%29%3Dr%5Bq%28x%29%5D)
![=r(2x+2)](https://tex.z-dn.net/?f=%3Dr%282x%2B2%29)
![=(2x+2)^2+1](https://tex.z-dn.net/?f=%3D%282x%2B2%29%5E2%2B1)
![=4x^2+8x+5](https://tex.z-dn.net/?f=%3D4x%5E2%2B8x%2B5)
Now, substituting x = -3, we get;
![(r \circ q)(-3)=4(-3)^2+8(-3)+5](https://tex.z-dn.net/?f=%28r%20%5Ccirc%20q%29%28-3%29%3D4%28-3%29%5E2%2B8%28-3%29%2B5)
![=4(9)-24+5](https://tex.z-dn.net/?f=%3D4%289%29-24%2B5)
![=36-24+5](https://tex.z-dn.net/?f=%3D36-24%2B5)
![(r \circ q)(-3)=17](https://tex.z-dn.net/?f=%28r%20%5Ccirc%20q%29%28-3%29%3D17)
Thus, the value of
is 17
<u>Value of </u>
<u>:</u>
Let us determine the value of ![(q \circ r)(x)](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28x%29)
![(q \circ r)(x)=q[r(x)]](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28x%29%3Dq%5Br%28x%29%5D)
![=q(x^2+1)](https://tex.z-dn.net/?f=%3Dq%28x%5E2%2B1%29)
![=2(x^2+1)+2](https://tex.z-dn.net/?f=%3D2%28x%5E2%2B1%29%2B2)
![=2x^2+2+2](https://tex.z-dn.net/?f=%3D2x%5E2%2B2%2B2)
![(q \circ r)(x)=2x^2+4](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28x%29%3D2x%5E2%2B4)
Substituting x = -3, we get;
![(q \circ r)(-3)=2(-3)^2+4](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28-3%29%3D2%28-3%29%5E2%2B4)
![=2(9)+4](https://tex.z-dn.net/?f=%3D2%289%29%2B4)
![(q \circ r)(-3)=22](https://tex.z-dn.net/?f=%28q%20%5Ccirc%20r%29%28-3%29%3D22)
Thus, the value of
is 22.
![\tan a=-\dfrac{12}5](https://tex.z-dn.net/?f=%5Ctan%20a%3D-%5Cdfrac%7B12%7D5)
Recall the following identities:
![1+\tan^2a=\sec^2a=\dfrac1{\cos^2a}](https://tex.z-dn.net/?f=1%2B%5Ctan%5E2a%3D%5Csec%5E2a%3D%5Cdfrac1%7B%5Ccos%5E2a%7D)
![\cos^2a=\dfrac{1+\cos2a}a](https://tex.z-dn.net/?f=%5Ccos%5E2a%3D%5Cdfrac%7B1%2B%5Ccos2a%7Da)
from which we get
![1+\left(-\dfrac{12}5\right)^2=\dfrac1{\cos^2a}](https://tex.z-dn.net/?f=1%2B%5Cleft%28-%5Cdfrac%7B12%7D5%5Cright%29%5E2%3D%5Cdfrac1%7B%5Ccos%5E2a%7D)
![\implies\cos^2a=\dfrac{25}{169}](https://tex.z-dn.net/?f=%5Cimplies%5Ccos%5E2a%3D%5Cdfrac%7B25%7D%7B169%7D)
Since
![a](https://tex.z-dn.net/?f=a)
is in the fourth quadrant
![\left(\dfrac{3\pi}2](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7B3%5Cpi%7D2%3Ca%3C2%5Cpi%5Cright%29)
we know that
![\cos a](https://tex.z-dn.net/?f=%5Ccos%20a)
should be positive, so when we take the square root here, we should take the positive root.
![\implies\cos a=\sqrt{\dfrac{25}{169}}=\dfrac5{13}](https://tex.z-dn.net/?f=%5Cimplies%5Ccos%20a%3D%5Csqrt%7B%5Cdfrac%7B25%7D%7B169%7D%7D%3D%5Cdfrac5%7B13%7D)
Now recall that
![\cos^2a+\sin^2a=1](https://tex.z-dn.net/?f=%5Ccos%5E2a%2B%5Csin%5E2a%3D1)
and since
![a](https://tex.z-dn.net/?f=a)
is in the fourth quadrant, we expect
![\sin a](https://tex.z-dn.net/?f=%5Csin%20a)
to be negative. So,
![\sin^2a=1-\cos^2a=\dfrac{144}{169}\implies\sin a=-\sqrt{\dfrac{144}{169}}=-\dfrac{12}{13}](https://tex.z-dn.net/?f=%5Csin%5E2a%3D1-%5Ccos%5E2a%3D%5Cdfrac%7B144%7D%7B169%7D%5Cimplies%5Csin%20a%3D-%5Csqrt%7B%5Cdfrac%7B144%7D%7B169%7D%7D%3D-%5Cdfrac%7B12%7D%7B13%7D)
One final identity:
![\sin2a=2\sin a\cos a](https://tex.z-dn.net/?f=%5Csin2a%3D2%5Csin%20a%5Ccos%20a)
from which we get
It’s a I think. I hope you get it right /)
There are 3 terms : 7x , 4y, and 1.
a term is separated by any math symbol
examples are =, +, *
A pendulum that swings fully once in 1.9 seconds has a length of 2.93 feet.
<h3>What is the length of a pendulum?</h3>
A basic pendulum is a machine in which the point mass is hung from a fixed support by a light, inextensible string.
Length of pendulum is defined as the distance between the point of suspension to the center of the bob and is denoted by " 1 ".
Given the equation:
![$$T=2 \Pi \sqrt{\frac{L}{32}}$$](https://tex.z-dn.net/?f=%24%24T%3D2%20%5CPi%20%5Csqrt%7B%5Cfrac%7BL%7D%7B32%7D%7D%24%24)
where L is the length in feet and T is the time in seconds.
Given that T = 1.9sec,
Hence:
![$$T=2 \Pi \sqrt{\frac{L}{32}}$$](https://tex.z-dn.net/?f=%24%24T%3D2%20%5CPi%20%5Csqrt%7B%5Cfrac%7BL%7D%7B32%7D%7D%24%24)
L = 2.93 feet
A pendulum that swings fully once in 1.9 sec has a length of 2.93 feet.
Learn more about Length of a pendulum here: brainly.com/question/8168512
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