50 as 1 in the ratio
50:200
50 as 4 in the ratio
50:12.5
hope this helps
I think it's x^4.
(Honestly, I feel 90% positive of this answer, so maybe just get one more opinion!)
I can’t see the “students work”. But the answer would be 250 words in 5 minutes.
Here’s how you would set up the equation:
25x(5*2)=
So your first step would be to multiply 5 and two because they’re in parenthesis, which would be 10.
So now you have:
25x10
25 times 10 is 250.
Hope that makes sense and helps! :)
Answer:
![\displaystyle \sin\left(2\sec^{-1}\left(\frac{u}{10}\right)\right)=\frac{20\sqrt{u^2-100}}{u^2}\text{ where } u>0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csin%5Cleft%282%5Csec%5E%7B-1%7D%5Cleft%28%5Cfrac%7Bu%7D%7B10%7D%5Cright%29%5Cright%29%3D%5Cfrac%7B20%5Csqrt%7Bu%5E2-100%7D%7D%7Bu%5E2%7D%5Ctext%7B%20where%20%7D%20u%3E0)
Step-by-step explanation:
We want to write the trignometric expression:
![\displaystyle \sin\left(2\sec^{-1}\left(\frac{u}{10}\right)\right)\text{ where } u>0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csin%5Cleft%282%5Csec%5E%7B-1%7D%5Cleft%28%5Cfrac%7Bu%7D%7B10%7D%5Cright%29%5Cright%29%5Ctext%7B%20where%20%7D%20u%3E0)
As an algebraic equation.
First, we can focus on the inner expression. Let θ equal the expression:
![\displaystyle \theta=\sec^{-1}\left(\frac{u}{10}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D%5Csec%5E%7B-1%7D%5Cleft%28%5Cfrac%7Bu%7D%7B10%7D%5Cright%29)
Take the secant of both sides:
![\displaystyle \sec(\theta)=\frac{u}{10}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csec%28%5Ctheta%29%3D%5Cfrac%7Bu%7D%7B10%7D)
Since secant is the ratio of the hypotenuse side to the adjacent side, this means that the opposite side is:
![\displaystyle o=\sqrt{u^2-10^2}=\sqrt{u^2-100}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20o%3D%5Csqrt%7Bu%5E2-10%5E2%7D%3D%5Csqrt%7Bu%5E2-100%7D)
By substitutition:
![\displaystyle= \sin(2\theta)](https://tex.z-dn.net/?f=%5Cdisplaystyle%3D%20%5Csin%282%5Ctheta%29)
Using an double-angle identity:
![=2\sin(\theta)\cos(\theta)](https://tex.z-dn.net/?f=%3D2%5Csin%28%5Ctheta%29%5Ccos%28%5Ctheta%29)
We know that the opposite side is √(u² -100), the adjacent side is 10, and the hypotenuse is u. Therefore:
![\displaystyle =2\left(\frac{\sqrt{u^2-100}}{u}\right)\left(\frac{10}{u}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D2%5Cleft%28%5Cfrac%7B%5Csqrt%7Bu%5E2-100%7D%7D%7Bu%7D%5Cright%29%5Cleft%28%5Cfrac%7B10%7D%7Bu%7D%5Cright%29)
Simplify. Therefore:
![\displaystyle \sin\left(2\sec^{-1}\left(\frac{u}{10}\right)\right)=\frac{20\sqrt{u^2-100}}{u^2}\text{ where } u>0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csin%5Cleft%282%5Csec%5E%7B-1%7D%5Cleft%28%5Cfrac%7Bu%7D%7B10%7D%5Cright%29%5Cright%29%3D%5Cfrac%7B20%5Csqrt%7Bu%5E2-100%7D%7D%7Bu%5E2%7D%5Ctext%7B%20where%20%7D%20u%3E0)