Diameter = 10.0002 inches, or 10 inches since were rounding to the nearest whole number.
Answer:
m∠N = 51°
m∠M = 31°
m∠O = 98°
Step-by-step explanation:
It is given that ΔMNO is an isosceles triangle with base NM.
m∠N = (4x + 7)° and m∠M = (2x + 29)°
By the property of an isosceles triangle,
Two legs of an isosceles triangle are equal in measure.
ON ≅ OM
And angles opposite to these equal sides measure the same.
m∠N = m∠M
(4x + 7) = (2x + 29)
4x - 2x = 29 - 7
2x = 22
x = 11
m∠N = (4x + 7)° = 51°
m∠M = (2x + 9)° = 31°
m∠O = 180° - (m∠N + m∠M)
= 180° - (51° + 31°)
= 180° - 82°
= 98°
We know, that the <span>area of the surface generated by revolving the curve y about the x-axis is given by:
![\boxed{A=2\pi\cdot\int\limits_a^by\sqrt{1+\left(y'\right)^2}\, dx}](https://tex.z-dn.net/?f=%5Cboxed%7BA%3D2%5Cpi%5Ccdot%5Cint%5Climits_a%5Eby%5Csqrt%7B1%2B%5Cleft%28y%27%5Cright%29%5E2%7D%5C%2C%20dx%7D)
In this case a = 0, b = 15,
![y=\dfrac{x^3}{15}](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7Bx%5E3%7D%7B15%7D)
and:
![y'=\left(\dfrac{x^3}{15}\right)'=\dfrac{3x^2}{15}=\boxed{\dfrac{x^2}{5}}](https://tex.z-dn.net/?f=y%27%3D%5Cleft%28%5Cdfrac%7Bx%5E3%7D%7B15%7D%5Cright%29%27%3D%5Cdfrac%7B3x%5E2%7D%7B15%7D%3D%5Cboxed%7B%5Cdfrac%7Bx%5E2%7D%7B5%7D%7D)
So there will be:
![A=2\pi\cdot\int\limits_0^{15}\dfrac{x^3}{15}\cdot\sqrt{1+\left(\dfrac{x^2}{5}\right)^2}\, dx=\dfrac{2\pi}{15}\cdot\int\limits_0^{15}x^3\cdot\sqrt{1+\dfrac{x^4}{25}}\,\, dx=\left(\star\right)\\\\-------------------------------\\\\ \int x^3\cdot\sqrt{1+\dfrac{x^4}{25}}\,\,dx=\int\sqrt{1+\dfrac{x^4}{25}}\cdot x^3\,dx=\left|\begin{array}{c}t=1+\dfrac{x^4}{25}\\\\dt=\dfrac{4x^3}{25}\,dx\\\\\dfrac{25}{4}\,dt=x^3\,dx\end{array}\right|=\\\\\\](https://tex.z-dn.net/?f=A%3D2%5Cpi%5Ccdot%5Cint%5Climits_0%5E%7B15%7D%5Cdfrac%7Bx%5E3%7D%7B15%7D%5Ccdot%5Csqrt%7B1%2B%5Cleft%28%5Cdfrac%7Bx%5E2%7D%7B5%7D%5Cright%29%5E2%7D%5C%2C%20dx%3D%5Cdfrac%7B2%5Cpi%7D%7B15%7D%5Ccdot%5Cint%5Climits_0%5E%7B15%7Dx%5E3%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%7D%5C%2C%5C%2C%20dx%3D%5Cleft%28%5Cstar%5Cright%29%5C%5C%5C%5C-------------------------------%5C%5C%5C%5C%0A%5Cint%20x%5E3%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%7D%5C%2C%5C%2Cdx%3D%5Cint%5Csqrt%7B1%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%7D%5Ccdot%20x%5E3%5C%2Cdx%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7Dt%3D1%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%5C%5C%5C%5Cdt%3D%5Cdfrac%7B4x%5E3%7D%7B25%7D%5C%2Cdx%5C%5C%5C%5C%5Cdfrac%7B25%7D%7B4%7D%5C%2Cdt%3Dx%5E3%5C%2Cdx%5Cend%7Barray%7D%5Cright%7C%3D%5C%5C%5C%5C%5C%5C)
![=\int\sqrt{t}\cdot\dfrac{25}{4}\,dt=\dfrac{25}{4}\int\sqrt{t}\,dt=\dfrac{25}{4}\int t^\frac{1}{2}\,dt=\dfrac{25}{4}\cdot\dfrac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}= \dfrac{25}{4}\cdot\dfrac{t^{\frac{3}{2}}}{\frac{3}{2}}=\\\\\\=\dfrac{25\cdot2}{4\cdot3}\,t^\frac{3}{2}=\boxed{\dfrac{25}{6}\,\left(1+\dfrac{x^4}{25}\right)^\frac{3}{2}}\\\\-------------------------------\\\\](https://tex.z-dn.net/?f=%3D%5Cint%5Csqrt%7Bt%7D%5Ccdot%5Cdfrac%7B25%7D%7B4%7D%5C%2Cdt%3D%5Cdfrac%7B25%7D%7B4%7D%5Cint%5Csqrt%7Bt%7D%5C%2Cdt%3D%5Cdfrac%7B25%7D%7B4%7D%5Cint%20t%5E%5Cfrac%7B1%7D%7B2%7D%5C%2Cdt%3D%5Cdfrac%7B25%7D%7B4%7D%5Ccdot%5Cdfrac%7Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%2B1%7D%7D%7B%5Cfrac%7B1%7D%7B2%7D%2B1%7D%3D%20%5Cdfrac%7B25%7D%7B4%7D%5Ccdot%5Cdfrac%7Bt%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B%5Cfrac%7B3%7D%7B2%7D%7D%3D%5C%5C%5C%5C%5C%5C%3D%5Cdfrac%7B25%5Ccdot2%7D%7B4%5Ccdot3%7D%5C%2Ct%5E%5Cfrac%7B3%7D%7B2%7D%3D%5Cboxed%7B%5Cdfrac%7B25%7D%7B6%7D%5C%2C%5Cleft%281%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%7D%5C%5C%5C%5C-------------------------------%5C%5C%5C%5C%20)
![\left(\star\right)=\dfrac{2\pi}{15}\cdot\int\limits_0^{15}x^3\cdot\sqrt{1+\dfrac{x^4}{25}}\,\, dx=\dfrac{2\pi}{15}\cdot\dfrac{25}{6}\cdot\left[\left(1+\dfrac{x^4}{25}\right)^\frac{3}{2}\right]_0^{15}=\\\\\\= \dfrac{5\pi}{9}\left[\left(1+\dfrac{15^4}{25}\right)^\frac{3}{2}-\left(1+\dfrac{0^4}{25}\right)^\frac{3}{2}\right]=\dfrac{5\pi}{9}\left[2026^\frac{3}{2}-1^\frac{3}{2}\right]=\\\\\\= \boxed{\dfrac{5\Big(2026^\frac{3}{2}-1\Big)}{9}\pi}](https://tex.z-dn.net/?f=%5Cleft%28%5Cstar%5Cright%29%3D%5Cdfrac%7B2%5Cpi%7D%7B15%7D%5Ccdot%5Cint%5Climits_0%5E%7B15%7Dx%5E3%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%7D%5C%2C%5C%2C%20dx%3D%5Cdfrac%7B2%5Cpi%7D%7B15%7D%5Ccdot%5Cdfrac%7B25%7D%7B6%7D%5Ccdot%5Cleft%5B%5Cleft%281%2B%5Cdfrac%7Bx%5E4%7D%7B25%7D%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cright%5D_0%5E%7B15%7D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Cdfrac%7B5%5Cpi%7D%7B9%7D%5Cleft%5B%5Cleft%281%2B%5Cdfrac%7B15%5E4%7D%7B25%7D%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D-%5Cleft%281%2B%5Cdfrac%7B0%5E4%7D%7B25%7D%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%3D%5Cdfrac%7B5%5Cpi%7D%7B9%7D%5Cleft%5B2026%5E%5Cfrac%7B3%7D%7B2%7D-1%5E%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Cboxed%7B%5Cdfrac%7B5%5CBig%282026%5E%5Cfrac%7B3%7D%7B2%7D-1%5CBig%29%7D%7B9%7D%5Cpi%7D)
Answer C.
</span>
Step-by-step explanation:
The answer is
X=23
Hope it helpsss