.....................hi.............
Answer:
59°
Step-by-step explanation:
First we find the value of
remembering that the angle that is formed in a straight line (in the diameter of the circle) is 180 °, so the sum :
![(11w)+(5w+4)](https://tex.z-dn.net/?f=%2811w%29%2B%285w%2B4%29)
must be equal to 180°:
![11w+5w+4=180](https://tex.z-dn.net/?f=11w%2B5w%2B4%3D180)
and we add the two terms with w:
![16w+4=180](https://tex.z-dn.net/?f=16w%2B4%3D180)
we move the +4 in the left as a -4 to the right side:
![16w=180-4\\16w=176](https://tex.z-dn.net/?f=16w%3D180-4%5C%5C16w%3D176)
and next we divide by 16:
![w=176/16\\w=11](https://tex.z-dn.net/?f=w%3D176%2F16%5C%5Cw%3D11)
And now that we know the value of
, we can find the measure of the arc YZ:
![YZ=5w+4](https://tex.z-dn.net/?f=YZ%3D5w%2B4)
substituting ![w=11](https://tex.z-dn.net/?f=w%3D11)
![YZ=5(11)+4\\YZ=55+4\\YZ=59](https://tex.z-dn.net/?f=YZ%3D5%2811%29%2B4%5C%5CYZ%3D55%2B4%5C%5CYZ%3D59)
The measure of YZ is 59°
![\dfrac{\mathrm d}{\mathrm dt}\left[2x^2+y^3\right]=\dfrac{\mathrm d}{\mathrm dt}[10]\implies 4x\dfrac{\mathrm dx}{\mathrm dt}+3y^2\dfrac{\mathrm dy}{\mathrm dt}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dt%7D%5Cleft%5B2x%5E2%2By%5E3%5Cright%5D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dt%7D%5B10%5D%5Cimplies%204x%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%2B3y%5E2%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%3D0)
When
![x=1](https://tex.z-dn.net/?f=x%3D1)
, the original equation tells you that you have
![2\times1^2+y^3=10\implies y^3=8\implies y=2](https://tex.z-dn.net/?f=2%5Ctimes1%5E2%2By%5E3%3D10%5Cimplies%20y%5E3%3D8%5Cimplies%20y%3D2)
You also know that when
![x=1](https://tex.z-dn.net/?f=x%3D1)
, you have
![\dfrac{\mathrm dy}{\mathrm dt}=3](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%3D3)
. Substituting everything you know into the differentiated equation, you get
![4\times1\times\dfrac{\mathrm dx}{\mathrm dt}+3\times2^2\times3=0\implies\dfrac{\mathrm dx}{\mathrm dt}=-9](https://tex.z-dn.net/?f=4%5Ctimes1%5Ctimes%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%2B3%5Ctimes2%5E2%5Ctimes3%3D0%5Cimplies%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%3D-9)
So
![x](https://tex.z-dn.net/?f=x)
is changing at a rate of
![-9\text{ units/min}](https://tex.z-dn.net/?f=-9%5Ctext%7B%20units%2Fmin%7D)
.