Answer:
I solved the question 1 to 14 but I'm sorry I cannot do 15 -20
Step-by-step explanation:
hope this helps
Answer: C) 4.20
Explanation:
Multiply each value with its corresponding frequency:
- 1*450 = 450
- 10*35 = 350
- 50*10 = 500
- 100*4 = 400
- 400*1 = 400
Add up the products
450+350+500+400+400 = 2100
Then divide this over the total frequency (450+35+10+4+1 = 500)
So we get 2100/500 = 4.20 as the expected value.
It depends on what you mean by the delimiting carats "^"...
Since you use parentheses appropriately in the answer choices, I'm going to go out on a limb here and assume something like "^x^" stands for
![\sqrt x](https://tex.z-dn.net/?f=%5Csqrt%20x)
.
In that case, you want to find the antiderivative,
![\displaystyle\int\frac{\mathrm dx}{\sqrt{9-8x-x^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Csqrt%7B9-8x-x%5E2%7D%7D)
Complete the square in the denominator:
![9-8x-x^2=25-(16+8x+x^2)=5^2-(x+4)^2](https://tex.z-dn.net/?f=9-8x-x%5E2%3D25-%2816%2B8x%2Bx%5E2%29%3D5%5E2-%28x%2B4%29%5E2)
Now substitute
![x+4=5\sin y](https://tex.z-dn.net/?f=x%2B4%3D5%5Csin%20y)
, so that
![\mathrm dx=5\cos y\,\mathrm dy](https://tex.z-dn.net/?f=%5Cmathrm%20dx%3D5%5Ccos%20y%5C%2C%5Cmathrm%20dy)
. Then
![\displaystyle\int\frac{\mathrm dx}{\sqrt{9-8x-x^2}}=\int\frac{5\cos y}{\sqrt{5^2-(5\sin y)^2}}\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Csqrt%7B9-8x-x%5E2%7D%7D%3D%5Cint%5Cfrac%7B5%5Ccos%20y%7D%7B%5Csqrt%7B5%5E2-%285%5Csin%20y%29%5E2%7D%7D%5C%2C%5Cmathrm%20dy)
which simplifies to
![\displaystyle\int\frac{5\cos y}{5\sqrt{1-\sin^2y}}\,\mathrm dy=\int\frac{\cos y}{\sqrt{\cos^2y}}\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B5%5Ccos%20%0Ay%7D%7B5%5Csqrt%7B1-%5Csin%5E2y%7D%7D%5C%2C%5Cmathrm%20dy%3D%5Cint%5Cfrac%7B%5Ccos%20y%7D%7B%5Csqrt%7B%5Ccos%5E2y%7D%7D%5C%2C%5Cmathrm%20dy)
Now, recall that
![\sqrt{x^2}=|x|](https://tex.z-dn.net/?f=%5Csqrt%7Bx%5E2%7D%3D%7Cx%7C)
. But we want the substitution we made to be reversible, so that
![x+4=5\sin y\iff y=\sin^{-1}\left(\dfrac{x+4}5\right)](https://tex.z-dn.net/?f=x%2B4%3D5%5Csin%20y%5Ciff%20y%3D%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%7Bx%2B4%7D5%5Cright%29)
which implies that
![-\dfrac\pi2\le y\le\dfrac\pi2](https://tex.z-dn.net/?f=-%5Cdfrac%5Cpi2%5Cle%20y%5Cle%5Cdfrac%5Cpi2)
. (This is the range of the inverse sine function.)
Under these conditions, we have
![\cos y\ge0](https://tex.z-dn.net/?f=%5Ccos%20y%5Cge0)
, which lets us reduce
![\sqrt{\cos^2y}=|\cos y|=\cos y](https://tex.z-dn.net/?f=%5Csqrt%7B%5Ccos%5E2y%7D%3D%7C%5Ccos%20y%7C%3D%5Ccos%20y)
. Finally,
![\displaystyle\int\frac{\cos y}{\cos y}\,\mathrm dy=\int\mathrm dy=y+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Ccos%20y%7D%7B%5Ccos%20y%7D%5C%2C%5Cmathrm%20dy%3D%5Cint%5Cmathrm%20dy%3Dy%2BC)
and back-substituting to get this in terms of
![x](https://tex.z-dn.net/?f=x)
yields
Each of the triangles are equal in base length, height, and area<span>. Remember the formula for the </span>area<span> of a triangle. The </span>area<span> of any triangle is 1/2 times the length of the base (which, in the </span>polygon<span>, is the length of a side) multiplied by the height (which is the same as the apothem in </span>regular polygon<span>).
</span>