It is 1.3333333333 repeated
For this case what you should know is that both functions are of the potential type.
We have then that
y = 2 * 2 ^ x This function grows exponentially upwards.
y = -2 * 5 ^ x This function grows exponentially downwards.
Answer See attached graphics.
Answer:
![AB = \sqrt{a^2 + b^2-2abCos\ C}](https://tex.z-dn.net/?f=AB%20%3D%20%5Csqrt%7Ba%5E2%20%2B%20b%5E2-2abCos%5C%20C%7D)
Step-by-step explanation:
Given:
The above triangle
Required
Solve for AB in terms of a, b and angle C
Considering right angled triangle BOC where O is the point between b-x and x
From BOC, we have that:
![Sin\ C = \frac{h}{a}](https://tex.z-dn.net/?f=Sin%5C%20C%20%3D%20%5Cfrac%7Bh%7D%7Ba%7D)
Make h the subject:
![h = aSin\ C](https://tex.z-dn.net/?f=h%20%3D%20aSin%5C%20C)
Also, in BOC (Using Pythagoras)
![a^2 = h^2 + x^2](https://tex.z-dn.net/?f=a%5E2%20%3D%20h%5E2%20%2B%20x%5E2)
Make
the subject
![x^2 = a^2 - h^2](https://tex.z-dn.net/?f=x%5E2%20%3D%20a%5E2%20-%20h%5E2)
Substitute
for h
becomes
![x^2 = a^2 - (aSin\ C)^2](https://tex.z-dn.net/?f=x%5E2%20%3D%20a%5E2%20-%20%28aSin%5C%20C%29%5E2)
![x^2 = a^2 - a^2Sin^2\ C](https://tex.z-dn.net/?f=x%5E2%20%3D%20a%5E2%20-%20a%5E2Sin%5E2%5C%20C)
Factorize
![x^2 = a^2 (1 - Sin^2\ C)](https://tex.z-dn.net/?f=x%5E2%20%3D%20a%5E2%20%281%20-%20Sin%5E2%5C%20C%29)
In trigonometry:
![Cos^2C = 1-Sin^2C](https://tex.z-dn.net/?f=Cos%5E2C%20%3D%201-Sin%5E2C)
So, we have that:
![x^2 = a^2 Cos^2\ C](https://tex.z-dn.net/?f=x%5E2%20%3D%20a%5E2%20Cos%5E2%5C%20C)
Take square roots of both sides
![x= aCos\ C](https://tex.z-dn.net/?f=x%3D%20aCos%5C%20C)
In triangle BOA, applying Pythagoras theorem, we have that:
![AB^2 = h^2 + (b-x)^2](https://tex.z-dn.net/?f=AB%5E2%20%3D%20h%5E2%20%2B%20%28b-x%29%5E2)
Open bracket
![AB^2 = h^2 + b^2-2bx+x^2](https://tex.z-dn.net/?f=AB%5E2%20%3D%20h%5E2%20%2B%20b%5E2-2bx%2Bx%5E2)
Substitute
and
in ![AB^2 = h^2 + b^2-2bx+x^2](https://tex.z-dn.net/?f=AB%5E2%20%3D%20h%5E2%20%2B%20b%5E2-2bx%2Bx%5E2)
![AB^2 = h^2 + b^2-2bx+x^2](https://tex.z-dn.net/?f=AB%5E2%20%3D%20h%5E2%20%2B%20b%5E2-2bx%2Bx%5E2)
![AB^2 = (aSin\ C)^2 + b^2-2b(aCos\ C)+(aCos\ C)^2](https://tex.z-dn.net/?f=AB%5E2%20%3D%20%28aSin%5C%20C%29%5E2%20%2B%20b%5E2-2b%28aCos%5C%20C%29%2B%28aCos%5C%20C%29%5E2)
Open Bracket
![AB^2 = a^2Sin^2\ C + b^2-2abCos\ C+a^2Cos^2\ C](https://tex.z-dn.net/?f=AB%5E2%20%3D%20a%5E2Sin%5E2%5C%20C%20%2B%20b%5E2-2abCos%5C%20C%2Ba%5E2Cos%5E2%5C%20C)
Reorder
![AB^2 = a^2Sin^2\ C +a^2Cos^2\ C + b^2-2abCos\ C](https://tex.z-dn.net/?f=AB%5E2%20%3D%20a%5E2Sin%5E2%5C%20C%20%2Ba%5E2Cos%5E2%5C%20C%20%2B%20b%5E2-2abCos%5C%20C)
Factorize:
![AB^2 = a^2(Sin^2\ C +Cos^2\ C) + b^2-2abCos\ C](https://tex.z-dn.net/?f=AB%5E2%20%3D%20a%5E2%28Sin%5E2%5C%20C%20%2BCos%5E2%5C%20C%29%20%2B%20b%5E2-2abCos%5C%20C)
In trigonometry:
![Sin^2C + Cos^2 = 1](https://tex.z-dn.net/?f=Sin%5E2C%20%2B%20Cos%5E2%20%3D%201)
So, we have that:
![AB^2 = a^2 * 1 + b^2-2abCos\ C](https://tex.z-dn.net/?f=AB%5E2%20%3D%20a%5E2%20%2A%201%20%2B%20b%5E2-2abCos%5C%20C)
![AB^2 = a^2 + b^2-2abCos\ C](https://tex.z-dn.net/?f=AB%5E2%20%3D%20a%5E2%20%2B%20b%5E2-2abCos%5C%20C)
Take square roots of both sides
![AB = \sqrt{a^2 + b^2-2abCos\ C}](https://tex.z-dn.net/?f=AB%20%3D%20%5Csqrt%7Ba%5E2%20%2B%20b%5E2-2abCos%5C%20C%7D)
Answer: y67
Step-by-step explanation:5n − 19 + n + 7 = 144 − 6n
6n − 12 = 144 − 6n
12n = 156
n = 13
m∠z = (144−6n)°
m∠z = (144−6×13)°
m∠z = y67