Remark
The point value is (-2,5) So we know the two sides. We need the hypotenuse. We should notice that the x value is minus (-2) and value is y value is plus (5). That means we are in quad 2. Be careful how you read that. (-2,5) is a point. It is not a tangent.
Step One
Find the hypotenuse.
a = - 2
b = 5
c = ??
c^2 = a^2 + b^2
c^2 = (-2)^2 + 5^2
c^2 = 4 + 25
c^2 = 29 Take the square root of both sides.
sqrt(c^2) = sqrt(29)
c = sqrt(29)
Step Two
Find the Cosine of the angle.
Cosine(theta) = adjacent / hypotenuse
Cosine(theta) = -2 / sqrt(29) <<<<<<< Answer
Again, watch out for what you are given.
An interesting question!
Instead of solving each set of non-linear system of equations 6 times, I will try a simpler way, which is more adapted to multiple choice questions. However, please be warned that this method will not improve immensely math skills, but will help with reasoning, and possibly a broader understand in system of equations.
For simplicity, I will denote the sets of system of non-linear equations by S={A,B,C,D,E,F} from left to right.
1. given solutions {(-2,3),(7,-6)}
We first check each member of S, i.e. A,B,C,D,E,F for the linear conditions.
A: x+y=-2+3=1 ≠ 3 so no
B: x-y=-2-3=-5 ≠ 1 so no
C: 2x+y=-2(2)+3=-1 ≠ 1 so no
D: x+2y=-2+3(2)=4 ≠ 2 so no
E: -x+y=2+3=5 ≠ 1 so no
F: x+y=-2+3=1 ≠ 3 so YES now check the other solution 7-6=-1 OK
Now check the non-linear condition for F:
S1: (-2,3)
y-15=3-15=-12
-x^2+4x=-(-2)^2+4*(-2)=-12 good
S2:(7,-6)
y-15=-6-15=-21
-x^2+4x=-(7^2)+4(7)=-49+28=-21 also good,
So Solution set (1) matches tile F
(2) Given solutions {(-5,8),(3,0)}
We proceed in a similar way, to find that
-5+8=3 & 3+0=3 (matches A) and nothing else.
Check non-linear conditions (optionally, I transposed the terms to make comparison easier)
A. x^2+x-y=(-5)^2+(-5)-8=25-5-8=12 (looks good)
x^2+x-y=(3)^2+(3)-0=9+3=12 (looks even better)
So we determined that {(-5,8),(3,0)} is the solution for tile A.
(3) For solution set {(-2,5),(3,-5)}
Check linear conditions:
x+y=3 and -3, so does not satisfy A,& F.
x-y=-7 and 8, so does not satisfy B
2x+y=5 and 1 so satisfies C
x+2y=8 & -7 so does not satisfy D
-x+y=7 & -8 so does not satisfy E
Thus C is our only set. NOTE: if two distinct points satisfy one linear condition (say C), then both points cannot satisfy another non-equivalent linear condition, i.e. we didn't really have to check points D & E (which are not equivalent linear conditions as C) once we have found C.
Now check non-linear conditions:
x^2-3x-y=(-2)^2-3(-2)-5=5 ok
x^2-3x-y=(3)^2-3(3)-(-5)=5 ok
So solution set (3) matches tile C.
@officiallyqueenz, for your benefit, I will leave one for you as exercise. Please proceed to find the system for solution set (4). Post a your work as comments if you get stuck, or your answer for verification if you wish.
For those who consider this answer as incomplete, please feel free to report the above answer. At his/her discretion, moderator may delete answer as incomplete, may delete the whole question as too many questions, or then again, may leave the answer intact for the better learning of the user who posed the question.
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Alex bought all the string needed for $125.
It costs $18 for the remaining materials to make each puppet.
So if we closely observe then we see that here $125 is the fixed cost because its not going to change with number of puppets.
And the variable cost is $18.
In this case we can model a Total cost function C(x) for for x number of puppets as below
The total cost to make 50 puppets=$1025
<h3>
Answer: B) 60</h3>
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Explanation:
Whenever the angle theta is between 0 and 90, the reference angle is exactly that value.
It's only when you get to other quadrants is when things get a bit tricky. Right now we're in quadrant 1, often written as Q1.
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Extra info:
- If theta is between 90 and 180, then the reference angle is 180-theta. This region is Q2
- If theta is in quadrant 3, between 180 and 270, then the reference angle is theta-180. The order of subtraction is important since x-y is the not the same as y-x.
- Lastly, if theta is between 270 and 360 (in Q4), then the reference angle is 360-theta.
- As you can see, we have four quadrants starting with Q1 in the upper right corner. Then we move counterclockwise to get Q2,Q3 and Q4.
