The correct answer is 64 bc 6 times 6 is 64
Answer:
The players ran 150 meters
Step-by-step explanation:
Pythagorean Theorem: a² + b² = c²
Since we have a rectangle with sides 90 and 120, we know if we split a diagonal across it, we will get a right triangle with legs 90 and 120. From there, we use Pythagorean Theorem to solve:
90² + 120² = c²
8100 + 14400 = c²
c² = 22500
√c² = √22500
c = 150
Answer:
![\displaystyle y'=3\frac{1+\frac{x}{\sqrt{1+x^2}}}{2+2x^2+2x\sqrt{1+x^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%3D3%5Cfrac%7B1%2B%5Cfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%7D%7B2%2B2x%5E2%2B2x%5Csqrt%7B1%2Bx%5E2%7D%7D)
Step-by-step explanation:
<u>The Derivative of a Function</u>
The derivative of f, also known as the instantaneous rate of change, or the slope of the tangent line to the graph of f, can be computed by the definition formula
![\displaystyle f'(x)=\lim\limits_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%3D%5Clim%5Climits_%7B%5CDelta%20x%20%5Crightarrow%200%7D%5Cfrac%7Bf%28x%2B%5CDelta%20x%29-f%28x%29%7D%7B%5CDelta%20x%7D)
There are tables where the derivative of all known functions are provided for an easy calculation of specific functions.
The derivative of the inverse tangent is given as
![\displaystyle (tan^{-1}u)'=\frac{u'}{1+u^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%28tan%5E%7B-1%7Du%29%27%3D%5Cfrac%7Bu%27%7D%7B1%2Bu%5E2%7D)
Where u is a function of x as provided:
![y=3tan^{-1}(x+\sqrt{1+x^2})](https://tex.z-dn.net/?f=y%3D3tan%5E%7B-1%7D%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%29)
If we set
![u=(x+\sqrt{1+x^2})](https://tex.z-dn.net/?f=u%3D%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%29)
Then
![\displaystyle u'=1+\frac{2x}{2\sqrt{1+x^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20u%27%3D1%2B%5Cfrac%7B2x%7D%7B2%5Csqrt%7B1%2Bx%5E2%7D%7D)
![\displaystyle u'=1+\frac{x}{\sqrt{1+x^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20u%27%3D1%2B%5Cfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D)
Taking the derivative of y
![y'=3[tan^{-1}(x+\sqrt{1+x^2})]'](https://tex.z-dn.net/?f=y%27%3D3%5Btan%5E%7B-1%7D%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%29%5D%27)
Using the change of variables
![\displaystyle y'=3[tan^{-1}u]'=3\frac{u'}{1+u^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%3D3%5Btan%5E%7B-1%7Du%5D%27%3D3%5Cfrac%7Bu%27%7D%7B1%2Bu%5E2%7D)
![\displaystyle y'=3\frac{u'}{1+u^2}=3\frac{1+\frac{x}{\sqrt{1+x^2}}}{1+(x+\sqrt{1+x^2})^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%3D3%5Cfrac%7Bu%27%7D%7B1%2Bu%5E2%7D%3D3%5Cfrac%7B1%2B%5Cfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%7D%7B1%2B%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%29%5E2%7D)
Operating
![\displaystyle y'=3\frac{1+\frac{x}{\sqrt{1+x^2}}}{1+x^2+2x\sqrt{1+x^2}+1+x^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%3D3%5Cfrac%7B1%2B%5Cfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%7D%7B1%2Bx%5E2%2B2x%5Csqrt%7B1%2Bx%5E2%7D%2B1%2Bx%5E2%7D)
![\boxed{\displaystyle y'=3\frac{1+\frac{x}{\sqrt{1+x^2}}}{2+2x^2+2x\sqrt{1+x^2}}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20y%27%3D3%5Cfrac%7B1%2B%5Cfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%7D%7B2%2B2x%5E2%2B2x%5Csqrt%7B1%2Bx%5E2%7D%7D%7D)
Answer
I am positive it us 4x = 14 I truly apologize if it's not.
Step-by-step explanation