Its the red one bc its between 1 and 12
Answer:
![f(x) =\sqrt{x} sin (x)](https://tex.z-dn.net/?f=%20f%28x%29%20%3D%5Csqrt%7Bx%7D%20sin%20%28x%29)
And on this case we can use the product rule for a derivate given by:
![\frac{d}{dx} (f(x)* g(x)) = f'(x) g(x) +f(x) g'(x)](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28f%28x%29%2A%20g%28x%29%29%20%3D%20f%27%28x%29%20g%28x%29%20%2Bf%28x%29%20g%27%28x%29)
Where
and
And replacing we have this:
![f'(x)= \frac{1}{2\sqrt{x}} sin (x) + \sqrt{x}cos(x)](https://tex.z-dn.net/?f=%20f%27%28x%29%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20sin%20%28x%29%20%2B%20%5Csqrt%7Bx%7Dcos%28x%29)
Step-by-step explanation:
We assume that the function of interest is:
![f(x) =\sqrt{x} sin (x)](https://tex.z-dn.net/?f=%20f%28x%29%20%3D%5Csqrt%7Bx%7D%20sin%20%28x%29)
And on this case we can use the product rule for a derivate given by:
![\frac{d}{dx} (f(x)* g(x)) = f'(x) g(x) +f(x) g'(x)](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28f%28x%29%2A%20g%28x%29%29%20%3D%20f%27%28x%29%20g%28x%29%20%2Bf%28x%29%20g%27%28x%29)
Where
and
And replacing we have this:
![f'(x)= \frac{1}{2\sqrt{x}} sin (x) + \sqrt{x}cos(x)](https://tex.z-dn.net/?f=%20f%27%28x%29%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20sin%20%28x%29%20%2B%20%5Csqrt%7Bx%7Dcos%28x%29)
Answer:
![V =\frac{1}{3} \pi r^2 h](https://tex.z-dn.net/?f=%20V%20%3D%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%20r%5E2%20h)
And replacing we got:
![V = \frac{1}{3} (3.14) (5in)^2 (2.5in) = 65.42 in^3](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20%283.14%29%20%285in%29%5E2%20%282.5in%29%20%3D%2065.42%20in%5E3)
And the best option would be:
65.42 Inches cubed
Step-by-step explanation:
For this case we have the following info given:
represent the height of the cone
represent the radius
And the volume of the cone is given by;
![V =\frac{1}{3} \pi r^2 h](https://tex.z-dn.net/?f=%20V%20%3D%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%20r%5E2%20h)
And replacing we got:
![V = \frac{1}{3} (3.14) (5in)^2 (2.5in) = 65.42 in^3](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20%283.14%29%20%285in%29%5E2%20%282.5in%29%20%3D%2065.42%20in%5E3)
And the best option would be:
65.42 Inches cubed
Answer:
30 and 47
their difference is 17 and when multiplied equal 1410
You can compute both the mean and second moment directly using the density function; in this case, it's
![f_X(x)=\begin{cases}\frac1{750-670}=\frac1{80}&\text{for }670\le x\le750\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_X%28x%29%3D%5Cbegin%7Bcases%7D%5Cfrac1%7B750-670%7D%3D%5Cfrac1%7B80%7D%26%5Ctext%7Bfor%20%7D670%5Cle%20x%5Cle750%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
Then the mean (first moment) is
![E[X]=\displaystyle\int_{-\infty}^\infty x\,f_X(x)\,\mathrm dx=\frac1{80}\int_{670}^{750}x\,\mathrm dx=710](https://tex.z-dn.net/?f=E%5BX%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20x%5C%2Cf_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac1%7B80%7D%5Cint_%7B670%7D%5E%7B750%7Dx%5C%2C%5Cmathrm%20dx%3D710)
and the second moment is
![E[X^2]=\displaystyle\int_{-\infty}^\infty x^2\,f_X(x)\,\mathrm dx=\frac1{80}\int_{670}^{750}x^2\,\mathrm dx=\frac{1,513,900}3](https://tex.z-dn.net/?f=E%5BX%5E2%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20x%5E2%5C%2Cf_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac1%7B80%7D%5Cint_%7B670%7D%5E%7B750%7Dx%5E2%5C%2C%5Cmathrm%20dx%3D%5Cfrac%7B1%2C513%2C900%7D3)
The second moment is useful in finding the variance, which is given by
![V[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2=\dfrac{1,513,900}3-710^2=\dfrac{1600}3](https://tex.z-dn.net/?f=V%5BX%5D%3DE%5B%28X-E%5BX%5D%29%5E2%5D%3DE%5BX%5E2%5D-E%5BX%5D%5E2%3D%5Cdfrac%7B1%2C513%2C900%7D3-710%5E2%3D%5Cdfrac%7B1600%7D3)
You get the standard deviation by taking the square root of the variance, and so
![\sqrt{V[X]}=\sqrt{\dfrac{1600}3}\approx23.09](https://tex.z-dn.net/?f=%5Csqrt%7BV%5BX%5D%7D%3D%5Csqrt%7B%5Cdfrac%7B1600%7D3%7D%5Capprox23.09)