How can you use the Distributive Property to solve multistep equations?
1 answer:
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<h3>Explanation:</h3>
1. "Create your own circle on a complex plane."
The equation of a circle in the complex plane can be written a number of ways. For center c (a complex number) and radius r (a positive real number), one formula is ...
|z-c| = r
If we let c = 2+i and r = 5, the equation becomes ...
|z -(2+i)| = 5
For z = x + yi and |z| = √(x² +y²), this equation is equivalent to the Cartesian coordinate equation ...
(x -2)² +(y -1)² = 5²
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2. "Choose two end points of a diameter to prove the diameter and radius of the circle."
We don't know what "prove the diameter and radius" means. We can show that the chosen end points z₁ and z₂ are 10 units apart, and their midpoint is the center of the circle c.
For the end points of a diameter, we choose ...
- z₁ = 5 +5i
- z₂ = -1 -3i
The distance between these is ...
|z₂ -z₁| = |(-1-5) +(-3-5)i| = |-6 -8i|
= √((-6)² +(-8)²) = √100
|z₂ -z₁| = 10 . . . . . . the diameter of a circle of radius 5
The midpoint of these two point should be the center of the circle.
(z₁ +z₂)/2 = ((5 -1) +(5 -3)i)/2 = (4 +2i)/2 = 2 +i
(z₁ +z₂)/2 = c . . . . . the center of the circle is the midpoint of the diameter
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3. "Show how to determine the center of the circle."
As with any circle, the center is the <em>midpoint of any diameter</em> (demonstrated in question 2). It is also the point of intersection of the perpendicular bisectors of any chords, and it is equidistant from any points on the circle.
Any of these relations can be used to find the circle center, depending on the information you start with.
As an example. we can choose another point we know to be on the circle:
z₄ = 6-2i
Using this point and the z₁ and z₂ above, we can write three equations in the "unknown" circle center (a +bi):
- |z₁ - (a+bi)| = r
- |z₂ - (a+bi)| = r
- |z₄ - (a+bi)| = r
Using the formula for the square of the magnitude of a complex number, this becomes ...
(5-a)² +(5-b)² = r² = 25 -10a +a² +25 -10b +b²
(-1-a)² +(-3-b)² = r² = 1 +2a +a² +9 +6b +b²
(6-a)² +(-2-b)² = r² = 36 -12a +a² +4 +4b +b²
Subtracting the first two equations from the third gives two linear equations in a and b:
11 -2a -21 +14b = 0
35 -14a -5 -2b = 0
Rearranging these to standard form, we get
a -7b = -5
7a +b = 15
Solving these by your favorite method gives ...
a +bi = 2 +i = c . . . . the center of the circle
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4. "Choose two points, one on the circle and the other not on the circle. Show, mathematically, how to determine whether or not the point is on the circle."
The points we choose are ...
- z₃ = 3 -2i
- z₄ = 6 -2i
We can show whether or not these are on the circle by seeing if they satisfy the equation of the circle.
|z -c| = 5
For z₃: |(3 -2i) -(2 +i)| = √((3-2)² +(-2-i)²) = √(1+9) = √10 ≠ 5 . . . NOT on circle
For z₄: |(6 -2i) -(2 +i)| = √((6 -2)² +(2 -i)²) = √(16 +9) = √25 = 5 . . . IS on circle
Answer:
a) S = {1, 2, 3}
b) P(odd number) =
c) No
d) Yes
Step-by-step explanation:
a) The sample space is the set of all possible outcomes. By definition, the elements of a set should not be repeated. Hence, the sample space S = {1, 2, 3}
However, the sample is not equiprobable because each element has different probabilities.
b) P(odd number) =
Note that the odd numbers are 1 (on three faces) and 3 (on one face).
c) The fact the die has been biased does not change the possible outcomes. It only changes the probability of getting any given number.
d) Because the 3-face has been loaded, this probability changes. In fact, it is calculated thus:
Let's assume the probability for 1 or 2 is . Then that of 3 is
(because it is twice the others). The sum of probabilities must be 1.
P(odd number) = Prob(1) + Prob(3)
= =