Bev has to cut her grandma's grass this weekend and wants to know exactly how much area she will be cutting. Calculate the area
of the polygon. Be sure to show all your work and explain your answer.
2 answers:
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Given <em>z</em> = 3 + <em>i</em>, right away we can find
(a) square
<em>z</em> ² = (3 + <em>i </em>)² = 3² + 6<em>i</em> + <em>i</em> ² = 9 + 6<em>i</em> - 1 = 8 + 6<em>i</em>
(b) modulus
|<em>z</em>| = √(3² + 1²) = √(9 + 1) = √10
(d) polar form
First find the argument:
arg(<em>z</em>) = arctan(1/3)
Then
<em>z</em> = |<em>z</em>| exp(<em>i</em> arg(<em>z</em>))
<em>z</em> = √10 exp(<em>i</em> arctan(1/3))
or
<em>z</em> = √10 (cos(arctan(1/3)) + <em>i</em> sin(arctan(1/3))
(c) square root
Any complex number has 2 square roots. Using the polar form from part (d), we have
√<em>z</em> = √(√10) exp(<em>i</em> arctan(1/3) / 2)
and
√<em>z</em> = √(√10) exp(<em>i</em> (arctan(1/3) + 2<em>π</em>) / 2)
Then in standard rectangular form, we have
and
We can simplify this further. We know that <em>z</em> lies in the first quadrant, so
0 < arg(<em>z</em>) = arctan(1/3) < <em>π</em>/2
which means
0 < 1/2 arctan(1/3) < <em>π</em>/4
Then both cos(1/2 arctan(1/3)) and sin(1/2 arctan(1/3)) are positive. Using the half-angle identity, we then have
and since cos(<em>x</em> + <em>π</em>) = -cos(<em>x</em>) and sin(<em>x</em> + <em>π</em>) = -sin(<em>x</em>),
Now, arctan(1/3) is an angle <em>y</em> such that tan(<em>y</em>) = 1/3. In a right triangle satisfying this relation, we would see that cos(<em>y</em>) = 3/√10 and sin(<em>y</em>) = 1/√10. Then
So the two square roots of <em>z</em> are
and
Using the normal distribution, it is found that the probability is 0.16.
<h3>Normal Probability Distribution</h3>In a normal distribution with mean and standard deviation
, the z-score of a measure X is given by:
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
In this problem, the mean and the standard deviation are given by, respectively, .
The proportion of students between 45 and 67 inches is the p-value of Z when <u>X = 67 subtracted by the p-value of Z when X = 45</u>, hence:
X = 67:
Z = -1
Z = -1 has a p-value of 0.16.
X = 45:
Z = -8.3
Z = -8.3 has a p-value of 0.
0.16 - 0 = 0.16
The probability is 0.16.
More can be learned about the normal distribution at brainly.com/question/24663213