Answer:
![5\sqrt{2}](https://tex.z-dn.net/?f=5%5Csqrt%7B2%7D)
Step-by-step explanation:
To rationalize the denominator, you need to multiply the numerator and denominator by the radical in the denominator (aka. a factor of 1):
![\frac{10}{\sqrt{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B%5Csqrt%7B2%7D%7D)
![\frac{10}{\sqrt{2}}*\frac{\sqrt{2}}{\sqrt{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B%5Csqrt%7B2%7D%7D%2A%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B2%7D%7D)
![\frac{10\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B10%5Csqrt%7B2%7D%7D%7B2%7D)
![5\sqrt{2}](https://tex.z-dn.net/?f=5%5Csqrt%7B2%7D)
Answer:
![s\approx17531](https://tex.z-dn.net/?f=s%5Capprox17531)
Step-by-step explanation:
When considering a curve defined by a function f(x) and its respective derivative f'(x) that are continuous in an interval [a, b], the length s of the arc delimited by a and b is given by the equation:
![s=\int\limits^a_b {\sqrt{1+(f'(x))^2} } \, dx](https://tex.z-dn.net/?f=s%3D%5Cint%5Climits%5Ea_b%20%7B%5Csqrt%7B1%2B%28f%27%28x%29%29%5E2%7D%20%7D%20%5C%2C%20dx)
First, let's find the derivate of y(x):
![y'(x)=3\sqrt{x}](https://tex.z-dn.net/?f=y%27%28x%29%3D3%5Csqrt%7Bx%7D)
Replacing the derivate in the previous equation:
![s=\int\limits^a_b {\sqrt{1+((3\sqrt{x}) ^2} } \, dx=\int\limits^a_b {\sqrt{1+9x} } \, dx](https://tex.z-dn.net/?f=s%3D%5Cint%5Climits%5Ea_b%20%7B%5Csqrt%7B1%2B%28%283%5Csqrt%7Bx%7D%29%20%5E2%7D%20%7D%20%5C%2C%20dx%3D%5Cint%5Climits%5Ea_b%20%7B%5Csqrt%7B1%2B9x%7D%20%7D%20%5C%2C%20dx)
Substitute u=1+9x and du=9x, so:
![s= \frac{1}{9} \int\limits^a_b u^{1/2} \, du](https://tex.z-dn.net/?f=s%3D%20%5Cfrac%7B1%7D%7B9%7D%20%5Cint%5Climits%5Ea_b%20u%5E%7B1%2F2%7D%20%20%5C%2C%20du)
The antiderivative of
is:
![s=\frac{2u^{3/2} }{27}](https://tex.z-dn.net/?f=s%3D%5Cfrac%7B2u%5E%7B3%2F2%7D%20%7D%7B27%7D)
u=1+9x, so:
![s=\frac{2(1+9x)^{3/2} }{27}](https://tex.z-dn.net/?f=s%3D%5Cfrac%7B2%281%2B9x%29%5E%7B3%2F2%7D%20%7D%7B27%7D)
Evaluating the antiderivative at the limits:
![s=\frac{2(1+9x)^{3/2} }{27} \left \{ {{b=432} \atop {a=31}} \right. =\frac{2(3889)^{3/2} }{27}-\frac{2(325)^{3/2} }{27} =17530.82988\approx17531](https://tex.z-dn.net/?f=s%3D%5Cfrac%7B2%281%2B9x%29%5E%7B3%2F2%7D%20%7D%7B27%7D%20%5Cleft%20%5C%7B%20%7B%7Bb%3D432%7D%20%5Catop%20%7Ba%3D31%7D%7D%20%5Cright.%20%3D%5Cfrac%7B2%283889%29%5E%7B3%2F2%7D%20%7D%7B27%7D-%5Cfrac%7B2%28325%29%5E%7B3%2F2%7D%20%7D%7B27%7D%20%3D17530.82988%5Capprox17531)
The output is 6, your current selected answer choice.
It would be positive 2, -3.