Find the slope of the line that passes through each pair of points. S (3, -6) and T (1, 4)
2 answers:
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Answer:
Step-by-step explanation:
Hello!
The variable of interest is
X: volume of root beer in a Windsor Bottling Company can.
A sample of n=24 cans was taken and their contents measured, resulting:
X[bar]= 11.4 oz
S= 0.62 oz
Assuming that the variable has a normal distribution X~N(μ;σ²), the parameter of interest is the average contents of the root beer cans of the Windsor Bottling Company (μ)
The claim is that the population mean content of the cans is different from 12 oz, symbolically: μ ≠ 12
The statistical hypothesis (Null and alternative) have to be complementary, exhaustive and mutually exclusive. The null hypothesis is the "no change" hypothesis and always carries the "=" sign.
If the claim is μ ≠ 12, its complement is μ = 12, the expression carrying the "=" sign will be the null hypothesis and its complement will be the alternative hypothesis:
H₀: μ = 12
H₁: μ ≠ 12
α: 0.05
To test the population mean of this normal population, you have to apply a one sample t-test, with statistic:
This test is two-tailed, using the critical value approach, you have to determine two rejection regions. Meaning, you'll reject the null hypothesis to small values of the statistic or to high values of the statistic.
The decision rule is:
If ≤ -2.069 or if
≥ 2.069, then you reject the null hypothesis.
If -2.069 < < 2.069, then you do not reject the null hypothesis.
The value is less than the left critical value, the decision is to reject the null hypothesis.
Then you can say that with a 5% significance level, there is significant evidence to reject the null hypothesis, then the average amount of root beer of the Windsor Bottling Company is different from 12 oz, this means that the claim about the amount of root beer in the cans is correct.
I hope it helps!
Answer:
5.25 m
Step-by-step explanation:
A diagram can help you understand the question, and can give you a clue as to how to find the answer. A diagram is attached. The problem can be described as finding the sum of two vectors whose magnitude and direction are known.
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<h3>understanding the direction</h3>In navigation problems, direction angles are specified a couple of different ways. A <em>bearing</em> is usually an angle in the range [0°, 360°), <em>measured clockwise from north</em>. In land surveying and some other applications, a bearing may be specified as an angle east or west of a north-south line. In this problem we are given the bearing of the second leg of the walk as ...
N 35° E . . . . . . . 35° east of north
Occasionally, a non-standard bearing will be given in terms of an angle north or south of an east-west line. The same bearing could be specified as E 55° N, for example.
<h3>the two vectors</h3>A vector is a mathematical object that has both magnitude and direction. It is sometimes expressed as an ordered pair: (magnitude; direction angle). It can also be expressed using some other notations;
- magnitude∠direction
- magnitude <em>cis</em> direction
In the latter case, "cis" is an abbreviation for the sum cos(θ)+i·sin(θ), where θ is the direction angle.
Sometimes a semicolon is used in the polar coordinate ordered pair to distinguish the coordinates from (x, y) rectangular coordinates.
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The first leg of the walk is 3 meters due north. The angle from north is 0°, and the magnitude of the distance is 3 meters. We can express this vector in any of the ways described above. One convenient way is 3∠0°.
The second leg of the walk is 2.5 meters on a bearing 35° clockwise from north. This leg can be described by the vector 2.5∠35°.
<h3>vector sum</h3>The final position is the sum of these two changes in position:
3∠0° +2.5∠35°
Some calculators can compute this sum directly. The result from one such calculator is shown in the second attachment:
= 5.24760∠15.8582°
This tells you the magnitude of the distance from the original position is about 5.25 meters. (This value is also shown in the first attachment.)
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You may have noticed that adding two vectors often results in a triangle. The magnitude of the vector sum can also be found using the Law of Cosines to solve the triangle. For the triangle shown in the first attachment, the Law of Cosines formula can be written as ...
a² = b² +o² -2bo·cos(A) . . . . where A is the internal angle at A, 145°
Using the values we know, this becomes ...
a² = 3² +2.5² -2(3)(2.5)cos(145°) ≈ 27.5373
a = √27.5373 = 5.24760 . . . . meters
The distance from the original position is about 5.25 meters.
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<em>Additional comment</em>
The vector sum can also be calculated in terms of rectangular coordinates. Position A has rectangular coordinates (0, 3). The change in coordinates from A to B can be represented as 2.5(sin(35°), cos(35°)) ≈ (1.434, 2.048). Then the coordinates of B are ...
(0, 3) +(1.434, 2.048) = (1.434, 5.048)
The distance can be found using the Pythagorean theorem:
OB = √(1.434² +5.048²) ≈ 5.248
