![ax-x^2=x(a-x)=0\implies x=0,x=a](https://tex.z-dn.net/?f=ax-x%5E2%3Dx%28a-x%29%3D0%5Cimplies%20x%3D0%2Cx%3Da)
which means the parabola
![y=ax-x^2](https://tex.z-dn.net/?f=y%3Dax-x%5E2)
intersects the x-axis at
![x=0](https://tex.z-dn.net/?f=x%3D0)
and
![x=a](https://tex.z-dn.net/?f=x%3Da)
. Assume
![a>0](https://tex.z-dn.net/?f=a%3E0)
.
Then the volume of the solid generated by revolving the region about the x-axis is
![\displaystyle\pi\int_0^a(ax-x^2)^2\,\mathrm dx=\dfrac{\pi a^5}{30}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cpi%5Cint_0%5Ea%28ax-x%5E2%29%5E2%5C%2C%5Cmathrm%20dx%3D%5Cdfrac%7B%5Cpi%20a%5E5%7D%7B30%7D)
Revolving about the y-axis, the volume would be
![\displaystyle2\pi\int_0^ax(ax-x^2)\,\mathrm dx=\dfrac{\pi a^4}6](https://tex.z-dn.net/?f=%5Cdisplaystyle2%5Cpi%5Cint_0%5Eax%28ax-x%5E2%29%5C%2C%5Cmathrm%20dx%3D%5Cdfrac%7B%5Cpi%20a%5E4%7D6)
The volumes are the same independent of which axis is taken as the axis of revolution, so
![\dfrac{\pi a^5}{30}=\dfrac{\pi a^4}6](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpi%20a%5E5%7D%7B30%7D%3D%5Cdfrac%7B%5Cpi%20a%5E4%7D6)
We're assuming
![a>0](https://tex.z-dn.net/?f=a%3E0)
, so we can safely divide both sides by
![\dfrac{\pi a^4}6](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpi%20a%5E4%7D6)
to get
![a=5](https://tex.z-dn.net/?f=a%3D5)
.
Answer:
vertex (–4,1), symmetry x=–4
Step-by-step explanation:
just use the formula
Answer:
The correct option is C
Step-by-step explanation:
Randomization is a method based on chance in which study participants are assigned to treatment groups.
Answer:
38.28cm^2
Step-by-step explanation:
Area of rectangle = 24cm^2
Area of rectangle = 8cm^2
Area of semicircle = 6.28cm^2
Okay. To find the upper quartile, first, line the numbers up in order. It would be this: 5, 7, 8, 10, 11, 13, 14, 18, 27. Then, we need to find the median. Cross the numbers out from furthest left to farthest right. When you do this, your median is 11. Now, ignore the median to find the quartile and repeat the same steps. You ignore the median whenever you have to solve for the quartiles. However, you’ll have to add the two numbers up and divide by 2 to find the quartiles in this case. 7 + 8 is 15. 15/2 = 7.5. The lower quartile is 7.5, but we’re looking for the upper quartile. Do the steps I said before to find it. 14 + 18 is 32. 32/2 is 16. There is your answer. The upper quartile is 16.