A)
![\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llccll} y = &{{ -1}}x^2&{{ +28}}x&{{ -192}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7B%20vertex%20of%20a%20vertical%20parabola%2C%20using%20coefficients%7D%5C%5C%5C%5C%0A%0A%5Cbegin%7Barray%7D%7Bllccll%7D%0Ay%20%3D%20%26%7B%7B%20-1%7D%7Dx%5E2%26%7B%7B%20%2B28%7D%7Dx%26%7B%7B%20-192%7D%7D%5C%5C%0A%26%5Cuparrow%20%26%5Cuparrow%20%26%5Cuparrow%20%5C%5C%0A%26a%26b%26c%0A%5Cend%7Barray%7D%5Cqquad%20%0A%5Cleft%28-%5Ccfrac%7B%7B%7B%20b%7D%7D%7D%7B2%7B%7B%20a%7D%7D%7D%5Cquad%20%2C%5Cquad%20%20%7B%7B%20c%7D%7D-%5Ccfrac%7B%7B%7B%20b%7D%7D%5E2%7D%7B4%7B%7B%20a%7D%7D%7D%5Cright%29)
check the picture below.
b)
![\bf f(x)=-x^2+28x-192\implies 0=-x^2+28x-192 \\\\\\ x^2-28x+192=0\implies (x-16)(x-12)=0\implies x= \begin{cases} 16\\ 12 \end{cases}](https://tex.z-dn.net/?f=%5Cbf%20f%28x%29%3D-x%5E2%2B28x-192%5Cimplies%200%3D-x%5E2%2B28x-192%0A%5C%5C%5C%5C%5C%5C%0Ax%5E2-28x%2B192%3D0%5Cimplies%20%28x-16%29%28x-12%29%3D0%5Cimplies%20x%3D%0A%5Cbegin%7Bcases%7D%0A16%5C%5C%0A12%0A%5Cend%7Bcases%7D)
what does this mean? check the picture below, notice the x-intercepts points.
There are three negative and two positive factors. The three negatives multiplied together would equal a negative number since 3 is odd. Then a negative multiplied by the remaining two positive factors would result in a negative answer.
Answer:
Step-by-step explanation:
A=1/36P2cot(20°)=1/36·1262·cot(20°)≈1211.63754in²
I hope this helped.!
Answer: 0.00459
Step-by-step explanation:
Answer:
The value of given expression is
.
Step-by-step explanation:
The given expression is
![\cos\left(\dfrac{19\pi}{12}\right)](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29)
The trigonometric ratios are not defined for
.
can be split into
.
![\cos\left(\dfrac{19\pi}{12}\right)=\cos (\frac{5\pi}{4}+\frac{\pi}{3})](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29%3D%5Ccos%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%29)
Using the addition formula
![\cos (A+B)=\cos A\cos B-\sin A\sin B](https://tex.z-dn.net/?f=%5Ccos%20%28A%2BB%29%3D%5Ccos%20A%5Ccos%20B-%5Csin%20A%5Csin%20B)
![\cos (\frac{5\pi}{4}+\frac{\pi}{3})=\cos( \frac{\pi}{3})\cdot \cos (\frac{5\pi}{4})-\sin( \frac{\pi}{3})\cdot \sin (\frac{5\pi}{4})](https://tex.z-dn.net/?f=%5Ccos%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%29%3D%5Ccos%28%20%5Cfrac%7B%5Cpi%7D%7B3%7D%29%5Ccdot%20%5Ccos%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29-%5Csin%28%20%5Cfrac%7B%5Cpi%7D%7B3%7D%29%5Ccdot%20%5Csin%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29)
We know that,
and ![\sin (\frac{\pi}{3})=\frac{\sqrt{3}}{2}](https://tex.z-dn.net/?f=%5Csin%20%28%5Cfrac%7B%5Cpi%7D%7B3%7D%29%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D)
![\cos\left(\dfrac{19\pi}{12}\right)=\frac{1}{2}\cdot \cos (\frac{5\pi}{4})-\frac{\sqrt{3}}{2}\cdot \sin (\frac{5\pi}{4})](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Ccos%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Ccdot%20%5Csin%20%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29)
lies in third quadrant, by using reference angle properties,
![\cos(\frac{5\pi}{4})=-\cos(\frac{\pi}{4})=-\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Ccos%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29%3D-%5Ccos%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![\sin(\frac{5\pi}{4})=-\sin(\frac{\pi}{4})=-\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Csin%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%29%3D-%5Csin%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![\cos\left(\dfrac{19\pi}{12}\right)=\frac{1}{2}\cdot (-\frac{\sqrt{2}}{2})-\frac{\sqrt{3}}{2}\cdot (-\frac{\sqrt{2}}{2})](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Ccdot%20%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29)
![\cos\left(\dfrac{19\pi}{12}\right)=-\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%2B%5Cfrac%7B%5Csqrt%7B6%7D%7D%7B4%7D)
![\cos\left(\dfrac{19\pi}{12}\right)=-\frac{(\sqrt{2}-\sqrt{6})}{4}](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B19%5Cpi%7D%7B12%7D%5Cright%29%3D-%5Cfrac%7B%28%5Csqrt%7B2%7D-%5Csqrt%7B6%7D%29%7D%7B4%7D)
Therefore the value of given expression is
.