Answer:
The red arrow shows the resultant vector. We have a Side Angle Side triangle ABC so can use The Cosine Rule:
a2=b2+c2−2bccosA
This becomes:
R2=7002+402−(2×700×40×cos45)
R2=491,600−39,597.9
R=672.3xkm/hr
This is the groundspeed of the aircraft.
To find θ we can use The Sine Rule:
sinCc=sinAa
This becomes:
sinθ40=sin45672.3
sinθ=0.04207
θ=2.41∘
This is known as the drift angle and is the correction the pilot should apply to remain on course.
The heading is the direction the aircraft's nose is pointing which is 000∘.
The track is the actual direction over the ground which is 357.6∘
An alternative method to this would be to separate each vector into vertical and horizontal components and add.
The resultant can be found using Pythagoras.
Answer:
19.4 degrees Celcius
Step-by-step explanation:
The formula of F to C is (F - 32) / 9 = C.
Answer:
Check below, please
Step-by-step explanation:
Hello!
1) In the Newton Method, we'll stop our approximations till the value gets repeated. Like this
![x_{1}=2\\x_{2}=2-\frac{f(2)}{f'(2)}=2.5\\x_{3}=2.5-\frac{f(2.5)}{f'(2.5)}\approx 2.4166\\x_{4}=2.4166-\frac{f(2.4166)}{f'(2.4166)}\approx 2.41421\\x_{5}=2.41421-\frac{f(2.41421)}{f'(2.41421)}\approx \mathbf{2.41421}](https://tex.z-dn.net/?f=x_%7B1%7D%3D2%5C%5Cx_%7B2%7D%3D2-%5Cfrac%7Bf%282%29%7D%7Bf%27%282%29%7D%3D2.5%5C%5Cx_%7B3%7D%3D2.5-%5Cfrac%7Bf%282.5%29%7D%7Bf%27%282.5%29%7D%5Capprox%202.4166%5C%5Cx_%7B4%7D%3D2.4166-%5Cfrac%7Bf%282.4166%29%7D%7Bf%27%282.4166%29%7D%5Capprox%202.41421%5C%5Cx_%7B5%7D%3D2.41421-%5Cfrac%7Bf%282.41421%29%7D%7Bf%27%282.41421%29%7D%5Capprox%20%5Cmathbf%7B2.41421%7D)
2) Looking at the graph, let's pick -1.2 and 3.2 as our approximations since it is a quadratic function. Passing through theses points -1.2 and 3.2 there are tangent lines that can be traced, which are the starting point to get to the roots.
We can rewrite it as: ![x^2-2x-4=0](https://tex.z-dn.net/?f=x%5E2-2x-4%3D0)
![x_{1}=-1.1\\x_{2}=-1.1-\frac{f(-1.1)}{f'(-1.1)}=-1.24047\\x_{3}=-1.24047-\frac{f(1.24047)}{f'(1.24047)}\approx -1.23607\\x_{4}=-1.23607-\frac{f(-1.23607)}{f'(-1.23607)}\approx -1.23606\\x_{5}=-1.23606-\frac{f(-1.23606)}{f'(-1.23606)}\approx \mathbf{-1.23606}](https://tex.z-dn.net/?f=x_%7B1%7D%3D-1.1%5C%5Cx_%7B2%7D%3D-1.1-%5Cfrac%7Bf%28-1.1%29%7D%7Bf%27%28-1.1%29%7D%3D-1.24047%5C%5Cx_%7B3%7D%3D-1.24047-%5Cfrac%7Bf%281.24047%29%7D%7Bf%27%281.24047%29%7D%5Capprox%20-1.23607%5C%5Cx_%7B4%7D%3D-1.23607-%5Cfrac%7Bf%28-1.23607%29%7D%7Bf%27%28-1.23607%29%7D%5Capprox%20-1.23606%5C%5Cx_%7B5%7D%3D-1.23606-%5Cfrac%7Bf%28-1.23606%29%7D%7Bf%27%28-1.23606%29%7D%5Capprox%20%5Cmathbf%7B-1.23606%7D)
As for
![x_{1}=3.2\\x_{2}=3.2-\frac{f(3.2)}{f'(3.2)}=3.23636\\x_{3}=3.23636-\frac{f(3.23636)}{f'(3.23636)}\approx 3.23606\\x_{4}=3.23606-\frac{f(3.23606)}{f'(3.23606)}\approx \mathbf{3.23606}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D3.2%5C%5Cx_%7B2%7D%3D3.2-%5Cfrac%7Bf%283.2%29%7D%7Bf%27%283.2%29%7D%3D3.23636%5C%5Cx_%7B3%7D%3D3.23636-%5Cfrac%7Bf%283.23636%29%7D%7Bf%27%283.23636%29%7D%5Capprox%203.23606%5C%5Cx_%7B4%7D%3D3.23606-%5Cfrac%7Bf%283.23606%29%7D%7Bf%27%283.23606%29%7D%5Capprox%20%5Cmathbf%7B3.23606%7D%5C%5C)
3) Rewriting and calculating its derivative. Remember to do it, in radians.
![5\cos(x)-x-1=0 \:and f'(x)=-5\sin(x)-1](https://tex.z-dn.net/?f=5%5Ccos%28x%29-x-1%3D0%20%5C%3Aand%20f%27%28x%29%3D-5%5Csin%28x%29-1)
![x_{1}=1\\x_{2}=1-\frac{f(1)}{f'(1)}=1.13471\\x_{3}=1.13471-\frac{f(1.13471)}{f'(1.13471)}\approx 1.13060\\x_{4}=1.13060-\frac{f(1.13060)}{f'(1.13060)}\approx 1.13059\\x_{5}= 1.13059-\frac{f( 1.13059)}{f'( 1.13059)}\approx \mathbf{ 1.13059}](https://tex.z-dn.net/?f=x_%7B1%7D%3D1%5C%5Cx_%7B2%7D%3D1-%5Cfrac%7Bf%281%29%7D%7Bf%27%281%29%7D%3D1.13471%5C%5Cx_%7B3%7D%3D1.13471-%5Cfrac%7Bf%281.13471%29%7D%7Bf%27%281.13471%29%7D%5Capprox%201.13060%5C%5Cx_%7B4%7D%3D1.13060-%5Cfrac%7Bf%281.13060%29%7D%7Bf%27%281.13060%29%7D%5Capprox%201.13059%5C%5Cx_%7B5%7D%3D%201.13059-%5Cfrac%7Bf%28%201.13059%29%7D%7Bf%27%28%201.13059%29%7D%5Capprox%20%5Cmathbf%7B%201.13059%7D)
For the second root, let's try -1.5
![x_{1}=-1.5\\x_{2}=-1.5-\frac{f(-1.5)}{f'(-1.5)}=-1.71409\\x_{3}=-1.71409-\frac{f(-1.71409)}{f'(-1.71409)}\approx -1.71410\\x_{4}=-1.71410-\frac{f(-1.71410)}{f'(-1.71410)}\approx \mathbf{-1.71410}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D-1.5%5C%5Cx_%7B2%7D%3D-1.5-%5Cfrac%7Bf%28-1.5%29%7D%7Bf%27%28-1.5%29%7D%3D-1.71409%5C%5Cx_%7B3%7D%3D-1.71409-%5Cfrac%7Bf%28-1.71409%29%7D%7Bf%27%28-1.71409%29%7D%5Capprox%20-1.71410%5C%5Cx_%7B4%7D%3D-1.71410-%5Cfrac%7Bf%28-1.71410%29%7D%7Bf%27%28-1.71410%29%7D%5Capprox%20%5Cmathbf%7B-1.71410%7D%5C%5C)
For x=-3.9, last root.
![x_{1}=-3.9\\x_{2}=-3.9-\frac{f(-3.9)}{f'(-3.9)}=-4.06438\\x_{3}=-4.06438-\frac{f(-4.06438)}{f'(-4.06438)}\approx -4.05507\\x_{4}=-4.05507-\frac{f(-4.05507)}{f'(-4.05507)}\approx \mathbf{-4.05507}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D-3.9%5C%5Cx_%7B2%7D%3D-3.9-%5Cfrac%7Bf%28-3.9%29%7D%7Bf%27%28-3.9%29%7D%3D-4.06438%5C%5Cx_%7B3%7D%3D-4.06438-%5Cfrac%7Bf%28-4.06438%29%7D%7Bf%27%28-4.06438%29%7D%5Capprox%20-4.05507%5C%5Cx_%7B4%7D%3D-4.05507-%5Cfrac%7Bf%28-4.05507%29%7D%7Bf%27%28-4.05507%29%7D%5Capprox%20%5Cmathbf%7B-4.05507%7D%5C%5C)
5) In this case, let's make a little adjustment on the Newton formula to find critical numbers. Remember their relation with 1st and 2nd derivatives.
![x_{n+1}=x_{n}-\frac{f'(n)}{f''(n)}](https://tex.z-dn.net/?f=x_%7Bn%2B1%7D%3Dx_%7Bn%7D-%5Cfrac%7Bf%27%28n%29%7D%7Bf%27%27%28n%29%7D)
![\mathbf{f'(x)=6x^5-4x^3+9x^2-2}](https://tex.z-dn.net/?f=%5Cmathbf%7Bf%27%28x%29%3D6x%5E5-4x%5E3%2B9x%5E2-2%7D)
![\mathbf{f''(x)=30x^4-12x^2+18x}](https://tex.z-dn.net/?f=%5Cmathbf%7Bf%27%27%28x%29%3D30x%5E4-12x%5E2%2B18x%7D)
For -1.2
![x_{1}=-1.2\\x_{2}=-1.2-\frac{f'(-1.2)}{f''(-1.2)}=-1.32611\\x_{3}=-1.32611-\frac{f'(-1.32611)}{f''(-1.32611)}\approx -1.29575\\x_{4}=-1.29575-\frac{f'(-1.29575)}{f''(-4.05507)}\approx -1.29325\\x_{5}= -1.29325-\frac{f'( -1.29325)}{f''( -1.29325)}\approx -1.29322\\x_{6}= -1.29322-\frac{f'( -1.29322)}{f''( -1.29322)}\approx \mathbf{-1.29322}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D-1.2%5C%5Cx_%7B2%7D%3D-1.2-%5Cfrac%7Bf%27%28-1.2%29%7D%7Bf%27%27%28-1.2%29%7D%3D-1.32611%5C%5Cx_%7B3%7D%3D-1.32611-%5Cfrac%7Bf%27%28-1.32611%29%7D%7Bf%27%27%28-1.32611%29%7D%5Capprox%20-1.29575%5C%5Cx_%7B4%7D%3D-1.29575-%5Cfrac%7Bf%27%28-1.29575%29%7D%7Bf%27%27%28-4.05507%29%7D%5Capprox%20-1.29325%5C%5Cx_%7B5%7D%3D%20-1.29325-%5Cfrac%7Bf%27%28%20-1.29325%29%7D%7Bf%27%27%28%20-1.29325%29%7D%5Capprox%20%20-1.29322%5C%5Cx_%7B6%7D%3D%20-1.29322-%5Cfrac%7Bf%27%28%20-1.29322%29%7D%7Bf%27%27%28%20-1.29322%29%7D%5Capprox%20%20%5Cmathbf%7B-1.29322%7D%5C%5C)
For x=0.4
![x_{1}=0.4\\x_{2}=0.4\frac{f'(0.4)}{f''(0.4)}=0.52476\\x_{3}=0.52476-\frac{f'(0.52476)}{f''(0.52476)}\approx 0.50823\\x_{4}=0.50823-\frac{f'(0.50823)}{f''(0.50823)}\approx 0.50785\\x_{5}= 0.50785-\frac{f'(0.50785)}{f''(0.50785)}\approx \mathbf{0.50785}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D0.4%5C%5Cx_%7B2%7D%3D0.4%5Cfrac%7Bf%27%280.4%29%7D%7Bf%27%27%280.4%29%7D%3D0.52476%5C%5Cx_%7B3%7D%3D0.52476-%5Cfrac%7Bf%27%280.52476%29%7D%7Bf%27%27%280.52476%29%7D%5Capprox%200.50823%5C%5Cx_%7B4%7D%3D0.50823-%5Cfrac%7Bf%27%280.50823%29%7D%7Bf%27%27%280.50823%29%7D%5Capprox%200.50785%5C%5Cx_%7B5%7D%3D%200.50785-%5Cfrac%7Bf%27%280.50785%29%7D%7Bf%27%27%280.50785%29%7D%5Capprox%20%20%5Cmathbf%7B0.50785%7D%5C%5C)
and for x=-0.4
![x_{1}=-0.4\\x_{2}=-0.4\frac{f'(-0.4)}{f''(-0.4)}=-0.44375\\x_{3}=-0.44375-\frac{f'(-0.44375)}{f''(-0.44375)}\approx -0.44173\\x_{4}=-0.44173-\frac{f'(-0.44173)}{f''(-0.44173)}\approx \mathbf{-0.44173}\\](https://tex.z-dn.net/?f=x_%7B1%7D%3D-0.4%5C%5Cx_%7B2%7D%3D-0.4%5Cfrac%7Bf%27%28-0.4%29%7D%7Bf%27%27%28-0.4%29%7D%3D-0.44375%5C%5Cx_%7B3%7D%3D-0.44375-%5Cfrac%7Bf%27%28-0.44375%29%7D%7Bf%27%27%28-0.44375%29%7D%5Capprox%20-0.44173%5C%5Cx_%7B4%7D%3D-0.44173-%5Cfrac%7Bf%27%28-0.44173%29%7D%7Bf%27%27%28-0.44173%29%7D%5Capprox%20%5Cmathbf%7B-0.44173%7D%5C%5C)
These roots (in bold) are the critical numbers
13 - 4/7 = <span>12.4285714286 Or when you round it 12.43</span>
Step 1: to solve this equation first subtract 4.9 from both sides.
w + 4.9 - 4.9 = 6.88 - 4.9
When you reduce that your answer should be w = 1.92
Therefore the weight of ingredients other than granola in Miranda's trail mix is 1.92