Answer:
Step-by-step explanation:
Our equations are
![y = -3x^2 + x + 12\\y = 2x^2 - 6x + 5\\y = x^2 + 7x - 11\\y = -x^2 - 8x - 16\\](https://tex.z-dn.net/?f=y%20%3D%20-3x%5E2%20%2B%20x%20%2B%2012%5C%5Cy%20%3D%202x%5E2%20-%206x%20%2B%205%5C%5Cy%20%3D%20x%5E2%20%2B%207x%20-%2011%5C%5Cy%20%3D%20-x%5E2%20-%208x%20-%2016%5C%5C)
Let us understand the term Discriminant of a quadratic equation and its properties
Discriminant is denoted by D and its formula is
![D=b^2-4ac\\](https://tex.z-dn.net/?f=D%3Db%5E2-4ac%5C%5C)
Where
a= the coefficient of the ![x^{2}](https://tex.z-dn.net/?f=x%5E%7B2%7D)
b= the coefficient of ![x](https://tex.z-dn.net/?f=x)
c = constant term
Properties of D: If D
i) D=0 , One real root
ii) D>0 , Two real roots
iii) D<0 , no real root
Hence in the given quadratic equations , we will find the values of D Discriminant and evaluate our answer accordingly .
Let us start with
![y = -3x^2 + x + 12\\a=-3 , b =1 , c =12\\D=1^2-4*(-3)*(12)\\D=1+144\\D=145\\D>0 \\](https://tex.z-dn.net/?f=y%20%3D%20-3x%5E2%20%2B%20x%20%2B%2012%5C%5Ca%3D-3%20%2C%20b%20%3D1%20%2C%20c%20%3D12%5C%5CD%3D1%5E2-4%2A%28-3%29%2A%2812%29%5C%5CD%3D1%2B144%5C%5CD%3D145%5C%5CD%3E0%20%5C%5C)
Hence we have two real roots for this equation.
![y = 2x^2 - 6x + 5\\](https://tex.z-dn.net/?f=y%20%3D%202x%5E2%20-%206x%20%2B%205%5C%5C)
![y = 2x^2 - 6x + 5\\a=2,b=-6,c=5\\D=(-6)^2-4*2*5\\D=36-40\\D=-4\\D](https://tex.z-dn.net/?f=y%20%3D%202x%5E2%20-%206x%20%2B%205%5C%5Ca%3D2%2Cb%3D-6%2Cc%3D5%5C%5CD%3D%28-6%29%5E2-4%2A2%2A5%5C%5CD%3D36-40%5C%5CD%3D-4%5C%5CD%3C0%5C%5C)
Hence we do not have any real root for this quadratic
![y = x^2 + 7x - 11\\a=1,b=7,-11\\D=7^2-4*1*(-11)\\D=49+44\\D=93\\](https://tex.z-dn.net/?f=y%20%3D%20x%5E2%20%2B%207x%20-%2011%5C%5Ca%3D1%2Cb%3D7%2C-11%5C%5CD%3D7%5E2-4%2A1%2A%28-11%29%5C%5CD%3D49%2B44%5C%5CD%3D93%5C%5C)
Hence D>0 and thus we have two real roots for this equation.
![y = -x^2 - 8x - 16\\a=-1,b=-8,c=-16\\D=(-8)^2-4*(-1)*(-16)\\D=64-64\\D=0\\](https://tex.z-dn.net/?f=y%20%3D%20-x%5E2%20-%208x%20-%2016%5C%5Ca%3D-1%2Cb%3D-8%2Cc%3D-16%5C%5CD%3D%28-8%29%5E2-4%2A%28-1%29%2A%28-16%29%5C%5CD%3D64-64%5C%5CD%3D0%5C%5C)
Hence we have one real root to this quadratic equation.
A correlation coefficient is always a value in between -1 and 1
The closest a coefficient to -1, the correlation is a strong negative correlation
The closest a coefficient to 1, the correlation is a strong positive correlation
The closest a coefficient to 0, there is no correlation at all
The coefficient -0.61 shows a strong negative correlation
This means that the relationship between the age and the violation is an inverse relationship; as age increases, violation decreases
Answer: option C
Answer:
Area of triangle is 25.
Step-by-step explanation:
We have been given an isosceles right triangle
Isosceles triangle is the triangle having two sides equal.
Figure is shown in attachment
By Pythagoras theorem
![BC^2=AC^2+AB^2](https://tex.z-dn.net/?f=BC%5E2%3DAC%5E2%2BAB%5E2)
AD is altitude which divides the triangle into two parts
DC=5 implies BC =10 since D equally divides BC
Let AC=a implies AB=a being Isosceles
On substituting the values in the Pythagoras theorem:
![10^2=a^2+a^2](https://tex.z-dn.net/?f=10%5E2%3Da%5E2%2Ba%5E2)
![100=2a^2](https://tex.z-dn.net/?f=100%3D2a%5E2)
![\Rightarrow a^2=50](https://tex.z-dn.net/?f=%5CRightarrow%20a%5E2%3D50)
![\Rightarrow a=\pm5\sqrt{2}](https://tex.z-dn.net/?f=%5CRightarrow%20a%3D%5Cpm5%5Csqrt%7B2%7D)
WE can find area of right triangle by considering height AB and AD
Area of triangle ABC is:
(1)
![\Rightarrow \frac{1}{2}\cdot 10\cdot AD](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%2010%5Ccdot%20AD)
And other method of area of triangle is:
(2)
Equating (1) and (2) we get:
![\frac{1}{2}\cdot 10\cdot AD=\frac{1}{2}\cdot a\cdot a](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ccdot%2010%5Ccdot%20AD%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20a%5Ccdot%20a)
![\Rightarrow AD=\frac{a^2}{10}](https://tex.z-dn.net/?f=%5CRightarrow%20AD%3D%5Cfrac%7Ba%5E2%7D%7B10%7D)
![\Rightarrow AD=\frac{50}{10}=5](https://tex.z-dn.net/?f=%5CRightarrow%20AD%3D%5Cfrac%7B50%7D%7B10%7D%3D5)
Using area of triangle is: ![\frac{1}{2}\cdot BC\cdot AD](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20BC%5Ccdot%20AD)
Now, the area of triangle ABC=![\frac{1}{2}\cdot 5\cdot 10](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ccdot%205%5Ccdot%2010)
![\Rightarrow 25](https://tex.z-dn.net/?f=%5CRightarrow%2025)
Answer:
![105.84\ cm^2](https://tex.z-dn.net/?f=105.84%5C%20cm%5E2)
Step-by-step explanation:
step 1
Find the area of the circle
The area of the circle is
![A=\pi r^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E%7B2%7D)
we have
----> the radius is half the diameter
substitute
![A=\pi (10.6)^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20%2810.6%29%5E%7B2%7D)
![A=112.36\pi\ cm^2](https://tex.z-dn.net/?f=A%3D112.36%5Cpi%5C%20cm%5E2)
step 2
Find the area of a sector
Remember that
The area of the circle subtends a central angle of 360 degrees
so
using proportion
Find out the area of a sector with a central angle of 108 degrees
![\frac{112.36\pi}{360^o}=\frac{x}{108^o}\\\\x=112.36\pi(108)/360\\\\x=33.708\pi\ cm^2](https://tex.z-dn.net/?f=%5Cfrac%7B112.36%5Cpi%7D%7B360%5Eo%7D%3D%5Cfrac%7Bx%7D%7B108%5Eo%7D%5C%5C%5C%5Cx%3D112.36%5Cpi%28108%29%2F360%5C%5C%5C%5Cx%3D33.708%5Cpi%5C%20cm%5E2)
assume
![\pi=3.14](https://tex.z-dn.net/?f=%5Cpi%3D3.14)
substitute
![33.708(3.14)=105.84\ cm^2](https://tex.z-dn.net/?f=33.708%283.14%29%3D105.84%5C%20cm%5E2)