Answer:X=6
Step-by-step explanation:
Step 1: Simplify both sides of the equation.
8x−2x−16=20
8x+−2x+−16=20
(8x+−2x)+(−16)=20(Combine Like Terms)
6x+−16=20
6x−16=20
Step 2: Add 16 to both sides.
6x−16+16=20+16
6x=36
Step 3: Divide both sides by 6.
6x
6
=
36
6
x=6
Answer:
x=118
y=37
Step-by-step explanation:
Since the lines are parallel:
Alternate Exterior angles are congruent, so x=118.
Linear pairs are supplementry, so y+25+118=180.
y+25=62
y=37.
(hope this helps and nice username :P)
I'll assume that what was meant was
![\sin ^4 x + \cos ^4 x = \dfrac{1}{2}](https://tex.z-dn.net/?f=%5Csin%20%5E4%20x%20%2B%20%5Ccos%20%5E4%20x%20%3D%20%5Cdfrac%7B1%7D%7B2%7D)
.
The exponent in the funny place is just an abbreviation:
![\sin ^4 x = (\sin x)^4](https://tex.z-dn.net/?f=%5Csin%20%5E4%20x%20%3D%20%28%5Csin%20x%29%5E4)
.
I hope that's what you meant. Let me know if I'm wrong.
Let's start from the old saw
![\cos^2 x + \sin ^2x = 1](https://tex.z-dn.net/?f=%5Ccos%5E2%20x%20%2B%20%5Csin%20%5E2x%20%3D%201)
Squaring both sides,
![(\cos^2 x + \sin ^2x)^2 = 1^2](https://tex.z-dn.net/?f=%28%5Ccos%5E2%20x%20%2B%20%5Csin%20%5E2x%29%5E2%20%3D%201%5E2)
![\cos^4 x + 2 \cos ^2 x \sin ^2x +\sin ^4x = 1](https://tex.z-dn.net/?f=%5Ccos%5E4%20x%20%2B%202%20%5Ccos%20%5E2%20x%20%5Csin%20%5E2x%20%2B%5Csin%20%5E4x%20%3D%201)
![\cos^4 x + \sin ^4x = 1 - 2 \cos ^2 x \sin ^2x](https://tex.z-dn.net/?f=%5Ccos%5E4%20x%20%2B%20%5Csin%20%5E4x%20%3D%201%20-%202%20%5Ccos%20%5E2%20x%20%5Csin%20%5E2x)
So now the original question
![\sin ^4 x + \cos ^4 x = \dfrac{1}{2}](https://tex.z-dn.net/?f=%5Csin%20%5E4%20x%20%2B%20%5Ccos%20%5E4%20x%20%3D%20%5Cdfrac%7B1%7D%7B2%7D)
becomes
![1 - 2 \cos ^2 x \sin ^2x = \dfrac{1}{2}](https://tex.z-dn.net/?f=%201%20-%202%20%5Ccos%20%5E2%20x%20%5Csin%20%5E2x%20%3D%20%5Cdfrac%7B1%7D%7B2%7D)
![4 \cos ^2 x \sin ^2x = 1](https://tex.z-dn.net/?f=%204%20%5Ccos%20%5E2%20x%20%5Csin%20%5E2x%20%3D%201)
Now we use the sine double angle formula
![\sin 2x = 2 \sin x \cos x](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%202%20%5Csin%20x%20%5Ccos%20x)
We square it to see
![\sin^2 2x = 4\sin^2 x \cos^2 x = 1](https://tex.z-dn.net/?f=%5Csin%5E2%202x%20%3D%204%5Csin%5E2%20x%20%5Ccos%5E2%20x%20%3D%201)
Taking the square root,
![\sin 2x = \pm 1](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%20%5Cpm%201)
Not sure how you want it; we'll do it in degrees.
When we know the sine of an angle, there's usually two angles on the unit circle that have that sine. They're supplementary angles which add to
![180^\circ](https://tex.z-dn.net/?f=180%5E%5Ccirc)
. But when the sine is 1 or -1 like it is here, we're looking at
![90^\circ](https://tex.z-dn.net/?f=90%5E%5Ccirc)
and
![-90^\circ](https://tex.z-dn.net/?f=-90%5E%5Ccirc)
, which are essentially their own supplements, slightly less messy.
That means we have two equations:
![\sin 2x = 1 = \sin 90^\circ](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%201%20%3D%20%5Csin%2090%5E%5Ccirc)
![2x = 90^\circ + 360^\circ k \quad](https://tex.z-dn.net/?f=2x%20%3D%2090%5E%5Ccirc%20%2B%20360%5E%5Ccirc%20k%20%5Cquad)
integer
![k](https://tex.z-dn.net/?f=k)
![x = 45^\circ + 180^\circ k](https://tex.z-dn.net/?f=x%20%3D%2045%5E%5Ccirc%20%2B%20180%5E%5Ccirc%20k)
or
![\sin 2x = -1 = \sin -90^\circ](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%20-1%20%3D%20%5Csin%20-90%5E%5Ccirc)
![2x = -90^\circ+ 360^\circ k](https://tex.z-dn.net/?f=2x%20%3D%20-90%5E%5Ccirc%2B%20360%5E%5Ccirc%20k)
![x = - 45^\circ + 180^\circ k](https://tex.z-dn.net/?f=x%20%3D%20-%2045%5E%5Ccirc%20%2B%20180%5E%5Ccirc%20k)
We can combine those for a final answer,
![x = \pm 45^\circ + 180^\circ k \quad](https://tex.z-dn.net/?f=x%20%3D%20%5Cpm%2045%5E%5Ccirc%20%2B%20180%5E%5Ccirc%20k%20%5Cquad)
integer
![k](https://tex.z-dn.net/?f=k)
Check. Let's just check one, how about
![x=-45^\circ + 180^\circ = 135^\circ](https://tex.z-dn.net/?f=x%3D-45%5E%5Ccirc%20%2B%20180%5E%5Ccirc%20%3D%20135%5E%5Ccirc)
![\sin(135)= 1/\sqrt{2}](https://tex.z-dn.net/?f=%5Csin%28135%29%3D%201%2F%5Csqrt%7B2%7D)
![\sin ^4(135)=(1/\sqrt{2})^4 = 1/4](https://tex.z-dn.net/?f=%5Csin%20%5E4%28135%29%3D%281%2F%5Csqrt%7B2%7D%29%5E4%20%3D%201%2F4)
![\cos ^4(135)=(-1/\sqrt{2})^4 = 1/4](https://tex.z-dn.net/?f=%5Ccos%20%5E4%28135%29%3D%28-1%2F%5Csqrt%7B2%7D%29%5E4%20%3D%201%2F4)
Answer: 0.40a + 0.40b
Step-by-step explanation:
Total Profit
= P(a,b) = ($0.40a) + ($0.40b),
where a = no. of apples sold and b = no. of bananas sold.
OK. So first, slope = rise/run. This can be calculated with any 2 points from the graph above.
I will use points (24, 168) and (8, 56).
Slope = (168 - 56) / (24 - 8)
Slope = 7
Since the y-intercept is 0, the equation is y = 7x
B should be the correct answer.
Cheers,
Brian