Answer:
rate of change = 1.5 dollars per mile
Step-by-step explanation:
cost for travelling 1 mile = 3.75
cost 2 miles = 5.25
cost 3 miles = 6.75
cost 4 miles = 8.25
Answer:
What is the question?
Step-by-step explanation:
Answer:
![A.\ \tan(x) \to 2.\ \sin(x) \sec(x)](https://tex.z-dn.net/?f=A.%5C%20%5Ctan%28x%29%20%5Cto%202.%5C%20%5Csin%28x%29%20%5Csec%28x%29)
![B.\ \cos(x) \to 5. \sec(x) - \sec(x)\sin^2(x)](https://tex.z-dn.net/?f=B.%5C%20%5Ccos%28x%29%20%5Cto%205.%20%5Csec%28x%29%20-%20%5Csec%28x%29%5Csin%5E2%28x%29)
![C.\ \sec(x)csc(x) \to 3. \tan(x) + \cot(x)](https://tex.z-dn.net/?f=C.%5C%20%5Csec%28x%29csc%28x%29%20%5Cto%203.%20%5Ctan%28x%29%20%2B%20%5Ccot%28x%29)
![D. \frac{1 - (cos(x))^2}{cos(x)} \to 1. \sin(x) \tan(x)](https://tex.z-dn.net/?f=D.%20%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D%20%5Cto%201.%20%5Csin%28x%29%20%5Ctan%28x%29)
![E.\ 2\sec(x) \to\ 4.\ \frac{\cos(x)}{1 - \sin(x)} +\frac{1-\sin(x)}{\cos(x)}](https://tex.z-dn.net/?f=E.%5C%202%5Csec%28x%29%20%5Cto%5C%204.%5C%20%5Cfrac%7B%5Ccos%28x%29%7D%7B1%20-%20%5Csin%28x%29%7D%20%2B%5Cfrac%7B1-%5Csin%28x%29%7D%7B%5Ccos%28x%29%7D)
Step-by-step explanation:
Given
![B.\ \cos(x)](https://tex.z-dn.net/?f=B.%5C%20%5Ccos%28x%29)
Required
Match the above with the appropriate identity from
![5.\ \sec(x) - \sec(x)(\sin(x))^2](https://tex.z-dn.net/?f=5.%5C%20%5Csec%28x%29%20-%20%5Csec%28x%29%28%5Csin%28x%29%29%5E2)
Solving (A):
![A.\ \tan(x)](https://tex.z-dn.net/?f=A.%5C%20%5Ctan%28x%29)
In trigonometry,
![\frac{sin(x)}{\cos(x)} = \tan(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bsin%28x%29%7D%7B%5Ccos%28x%29%7D%20%3D%20%5Ctan%28x%29)
So, we have:
![\tan(x) = \frac{\sin(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Ctan%28x%29%20%3D%20%5Cfrac%7B%5Csin%28x%29%7D%7B%5Ccos%28x%29%7D)
Split
![\tan(x) = \sin(x) * \frac{1}{\cos(x)}](https://tex.z-dn.net/?f=%5Ctan%28x%29%20%3D%20%5Csin%28x%29%20%2A%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D)
In trigonometry
![\frac{1}{\cos(x)} =sec(x)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%20%3Dsec%28x%29)
So, we have:
![\tan(x) = \sin(x) * \sec(x)](https://tex.z-dn.net/?f=%5Ctan%28x%29%20%3D%20%5Csin%28x%29%20%2A%20%5Csec%28x%29)
--- proved
Solving (b):
![B.\ \cos(x)](https://tex.z-dn.net/?f=B.%5C%20%5Ccos%28x%29)
Multiply by
--- an equivalent of 1
So, we have:
![\cos(x) = \cos(x) * \frac{\cos(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Ccos%28x%29%20%2A%20%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Ccos%28x%29%7D)
![\cos(x) = \frac{\cos^2(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%7D%7B%5Ccos%28x%29%7D)
In trigonometry:
![\cos^2(x) = 1 - \sin^2(x)](https://tex.z-dn.net/?f=%5Ccos%5E2%28x%29%20%3D%201%20-%20%5Csin%5E2%28x%29)
So, we have:
![\cos(x) = \frac{1 - \sin^2(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Cfrac%7B1%20-%20%5Csin%5E2%28x%29%7D%7B%5Ccos%28x%29%7D)
Split
![\cos(x) = \frac{1}{\cos(x)} - \frac{\sin^2(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%20-%20%5Cfrac%7B%5Csin%5E2%28x%29%7D%7B%5Ccos%28x%29%7D)
Rewrite as:
![\cos(x) = \frac{1}{\cos(x)} - \frac{1}{\cos(x)}*\sin^2(x)](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%20-%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%2A%5Csin%5E2%28x%29)
Express ![\frac{1}{\cos(x)}\ as\ \sec(x)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%5C%20as%5C%20%5Csec%28x%29)
![\cos(x) = \sec(x) - \sec(x) * \sin^2(x)](https://tex.z-dn.net/?f=%5Ccos%28x%29%20%3D%20%5Csec%28x%29%20-%20%5Csec%28x%29%20%2A%20%5Csin%5E2%28x%29)
--- proved
Solving (C):
![C.\ \sec(x)csc(x)](https://tex.z-dn.net/?f=C.%5C%20%5Csec%28x%29csc%28x%29)
In trigonometry
![\sec(x)= \frac{1}{\cos(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D)
and
![\csc(x)= \frac{1}{\sin(x)}](https://tex.z-dn.net/?f=%5Ccsc%28x%29%3D%20%5Cfrac%7B1%7D%7B%5Csin%28x%29%7D)
So, we have:
![\sec(x)csc(x) = \frac{1}{\cos(x)}*\frac{1}{\sin(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%2A%5Cfrac%7B1%7D%7B%5Csin%28x%29%7D)
Multiply by
--- an equivalent of 1
![\sec(x)csc(x) = \frac{1}{\cos(x)}*\frac{1}{\sin(x)} * \frac{\cos(x)}{\cos(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D%2A%5Cfrac%7B1%7D%7B%5Csin%28x%29%7D%20%2A%20%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Ccos%28x%29%7D)
![\sec(x)csc(x) = \frac{1}{\cos^2(x)}*\frac{\cos(x)}{\sin(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%5E2%28x%29%7D%2A%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Csin%28x%29%7D)
Express
and ![\frac{\cos(x)}{\sin(x)}\ as\ \frac{1}{\tan(x)}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Csin%28x%29%7D%5C%20as%5C%20%5Cfrac%7B1%7D%7B%5Ctan%28x%29%7D)
![\sec(x)csc(x) = \sec^2(x)*\frac{1}{\tan(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Csec%5E2%28x%29%2A%5Cfrac%7B1%7D%7B%5Ctan%28x%29%7D)
![\sec(x)csc(x) = \frac{\sec^2(x)}{\tan(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B%5Csec%5E2%28x%29%7D%7B%5Ctan%28x%29%7D)
In trigonometry:
![tan^2(x) + 1 =\sec^2(x)](https://tex.z-dn.net/?f=tan%5E2%28x%29%20%2B%201%20%3D%5Csec%5E2%28x%29)
So, we have:
![\sec(x)csc(x) = \frac{\tan^2(x) + 1}{\tan(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B%5Ctan%5E2%28x%29%20%2B%201%7D%7B%5Ctan%28x%29%7D)
Split
![\sec(x)csc(x) = \frac{\tan^2(x)}{\tan(x)} + \frac{1}{\tan(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29csc%28x%29%20%3D%20%5Cfrac%7B%5Ctan%5E2%28x%29%7D%7B%5Ctan%28x%29%7D%20%2B%20%5Cfrac%7B1%7D%7B%5Ctan%28x%29%7D)
Simplify
proved
Solving (D)
![D.\ \frac{1 - (cos(x))^2}{cos(x)}](https://tex.z-dn.net/?f=D.%5C%20%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D)
Open bracket
![\frac{1 - (cos(x))^2}{cos(x)} = \frac{1 - cos^2(x)}{cos(x)}](https://tex.z-dn.net/?f=%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D%20%3D%20%5Cfrac%7B1%20-%20cos%5E2%28x%29%7D%7Bcos%28x%29%7D)
![1 - \cos^2(x) = \sin^2(x)](https://tex.z-dn.net/?f=1%20-%20%5Ccos%5E2%28x%29%20%3D%20%5Csin%5E2%28x%29)
So, we have:
![\frac{1 - (cos(x))^2}{cos(x)} = \frac{sin^2(x)}{cos(x)}](https://tex.z-dn.net/?f=%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D%20%3D%20%5Cfrac%7Bsin%5E2%28x%29%7D%7Bcos%28x%29%7D)
Split
![\frac{1 - (cos(x))^2}{cos(x)} = \sin(x) * \frac{sin(x)}{cos(x)}](https://tex.z-dn.net/?f=%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D%20%3D%20%5Csin%28x%29%20%2A%20%5Cfrac%7Bsin%28x%29%7D%7Bcos%28x%29%7D)
![\frac{sin(x)}{\cos(x)} = \tan(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bsin%28x%29%7D%7B%5Ccos%28x%29%7D%20%3D%20%5Ctan%28x%29)
So, we have:
![\frac{1 - (cos(x))^2}{cos(x)} = \sin(x) * \tan(x)](https://tex.z-dn.net/?f=%5Cfrac%7B1%20-%20%28cos%28x%29%29%5E2%7D%7Bcos%28x%29%7D%20%3D%20%5Csin%28x%29%20%2A%20%5Ctan%28x%29)
--- proved
Solving (E):
![E.\ 2\sec(x)](https://tex.z-dn.net/?f=E.%5C%202%5Csec%28x%29)
In trigonometry
![\sec(x)= \frac{1}{\cos(x)}](https://tex.z-dn.net/?f=%5Csec%28x%29%3D%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D)
So, we have:
![2\sec(x) = 2 * \frac{1}{\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%202%20%2A%20%5Cfrac%7B1%7D%7B%5Ccos%28x%29%7D)
![2\sec(x) = \frac{2}{\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B2%7D%7B%5Ccos%28x%29%7D)
Multiply by
--- an equivalent of 1
![2\sec(x) = \frac{2}{\cos(x)} * \frac{1 - \sin(x)}{1 - \sin(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B2%7D%7B%5Ccos%28x%29%7D%20%2A%20%5Cfrac%7B1%20-%20%5Csin%28x%29%7D%7B1%20-%20%5Csin%28x%29%7D)
![2\sec(x) = \frac{2(1 - \sin(x))}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B2%281%20-%20%5Csin%28x%29%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Open bracket
![2\sec(x) = \frac{2 - 2\sin(x)}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B2%20-%202%5Csin%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Express 2 as 1 + 1
![2\sec(x) = \frac{1+1 - 2\sin(x)}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B1%2B1%20-%202%5Csin%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Express 1 as ![\sin^2(x) + \cos^2(x)](https://tex.z-dn.net/?f=%5Csin%5E2%28x%29%20%2B%20%5Ccos%5E2%28x%29)
![2\sec(x) = \frac{\sin^2(x) + \cos^2(x)+1 - 2\sin(x)}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Csin%5E2%28x%29%20%2B%20%5Ccos%5E2%28x%29%2B1%20-%202%5Csin%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Rewrite as:
![2\sec(x) = \frac{\cos^2(x)+1 - 2\sin(x)+\sin^2(x)}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%2B1%20-%202%5Csin%28x%29%2B%5Csin%5E2%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Expand
![2\sec(x) = \frac{\cos^2(x)+1 - \sin(x)- \sin(x)+\sin^2(x)}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%2B1%20-%20%5Csin%28x%29-%20%5Csin%28x%29%2B%5Csin%5E2%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Factorize
![2\sec(x) = \frac{\cos^2(x)+1(1 - \sin(x))- \sin(x)(1-\sin(x))}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%2B1%281%20-%20%5Csin%28x%29%29-%20%5Csin%28x%29%281-%5Csin%28x%29%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Factor out 1 - sin(x)
![2\sec(x) = \frac{\cos^2(x)+(1- \sin(x))(1-\sin(x))}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%2B%281-%20%5Csin%28x%29%29%281-%5Csin%28x%29%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Express as squares
![2\sec(x) = \frac{\cos^2(x)+(1-\sin(x))^2}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%2B%281-%5Csin%28x%29%29%5E2%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Split
![2\sec(x) = \frac{\cos^2(x)}{(1 - \sin(x))\cos(x)} +\frac{(1-\sin(x))^2}{(1 - \sin(x))\cos(x)}](https://tex.z-dn.net/?f=2%5Csec%28x%29%20%3D%20%5Cfrac%7B%5Ccos%5E2%28x%29%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D%20%2B%5Cfrac%7B%281-%5Csin%28x%29%29%5E2%7D%7B%281%20-%20%5Csin%28x%29%29%5Ccos%28x%29%7D)
Cancel out like factors
--- proved
Solution 1:
In the given statement :
"The settlers traveled west in covered wagons"
The Predicate is <u> traveled west in covered wagons.</u>
So option c. traveled west in covered wagons is the correct choice.
Solution 2:
In the given sentence:
"Soon, telegraph lines provided additional communication to the city."
The simple subject and verb are given by lines provided.
Answer : Option a: lines provided.