Answer:
The answer to your question is: letter B
Step-by-step explanation:
Data
A (-5, 4)
B (4, 1)
Formula
slope = m = ![\frac{1 - 4}{4 + 5}](https://tex.z-dn.net/?f=%5Cfrac%7B1%20-%204%7D%7B4%20%2B%205%7D)
m =
= ![\frac{-1}{ 3}](https://tex.z-dn.net/?f=%5Cfrac%7B-1%7D%7B%203%7D)
Point slope
(y - 4) =
(x + 5)
Answer:
9/4
Step-by-step explanation:
x/6 = 3/8
or, 8x = 3*6
or, x = 18/8
or, x = 9/4
Answer:
2
Step-by-step explanation:
There are three numbers on the first spinner. Two are not even (1 and 3).
So there are 2 possible outcomes that work: 1 and 6 or 3 and 6.
Given that
![r=\theta](https://tex.z-dn.net/?f=r%3D%5Ctheta)
, then
![r'=1](https://tex.z-dn.net/?f=r%27%3D1)
The slope of a tangent line in the polar coordinate is given by:
![m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}](https://tex.z-dn.net/?f=m%3D%20%5Cfrac%7Br%27%5Csin%5Ctheta%2Br%5Ccos%5Ctheta%7D%7Br%27%5Ccos%5Ctheta-r%5Csin%5Ctheta%7D%20)
Thus, we have:
![m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta}](https://tex.z-dn.net/?f=m%3D%20%5Cfrac%7B%5Csin%5Ctheta%2B%5Ctheta%5Ccos%5Ctheta%7D%7B%5Ccos%5Ctheta-%5Ctheta%5Csin%5Ctheta%7D%20)
Part A:
For horizontal tangent lines, m = 0.
Thus, we have:
![\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta](https://tex.z-dn.net/?f=%5Csin%5Ctheta%2B%5Ctheta%5Ccos%5Ctheta%3D0%20%5C%5C%20%20%5C%5C%20%5Ctheta%5Ccos%5Ctheta%3D-%5Csin%5Ctheta%20%5C%5C%20%20%5C%5C%20%5Ctheta%3D-%20%5Cfrac%7B%5Csin%5Ctheta%7D%7B%5Ccos%5Ctheta%7D%20%3D-%5Ctan%5Ctheta)
Therefore, the <span>values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:
</span><span>θ = 0
</span>θ = <span>2.02875783811043
</span>
θ = <span>4.91318043943488
Part B:
For vertical tangent lines,
![\frac{1}{m} =0](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bm%7D%20%3D0)
Thus, we have:
![\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta](https://tex.z-dn.net/?f=%5Ccos%5Ctheta-%5Ctheta%5Csin%5Ctheta%3D0%20%5C%5C%20%20%5C%5C%20%5CRightarrow%5Ctheta%5Csin%5Ctheta%3D%5Ccos%5Ctheta%20%5C%5C%20%20%5C%5C%20%5CRightarrow%5Ctheta%3D%20%5Cfrac%7B%5Ccos%5Ctheta%7D%7B%5Csin%5Ctheta%7D%20%3D%5Csec%5Ctheta)
</span>Therefore, the <span>values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:
</span>θ = <span>4.91718592528713</span>