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3 years ago

Describe the geometric meaning of the following mappings in spherical coordinates.

Mathematics

1 answer:

bearhunter [10]3 years ago

8 0

Answer:

A. (iv)-There is a rotation by π around the x-axis.

B. (v)-There is a reflection across the xy plane.

C. None of the above, There is a rotation by π/2 about the x-axis with a radial expansion by a factor of 2.

Step-by-step explanation:

A. In spherical coordinates, for some point in R³, the angle θ is designed to be the angle between the x-axis and the projection of the line that goes from (0,0,0) to that point into the xy plane.

So, if you add π to the angle it only rotate 180 degrees or π units about the x-axis

B. In spherical coordinates, for some point in R³, the angle ϕ is designed to be the angle between the z-axis and the diagonal from (0,0,0) to that point.

So, if you subtract π to the angle it would flip the diagonal in the negative part of z (assuming ϕ<π/2, otherwise, originally the diagonal would be in the negative part of z and it would flip to the positive part of z) and this could be interpreted as a reflection around the xy plane. Notice if ϕ=π/2, then π − ϕ=π/2 so everything stays the same for points that belong to xy plane (i.e that ϕ=π/2)

C. In spherical coordinates, for some point in R³, ρ is designed to be the length of the diagonal from (0,0,0) to that point. So if we multiply ρ by 2, we are expanding the length of the diagonal.

But as we´ve seen before, adding something to θ means a rotation about x-axis by what we are adding (π/2)

Then the answer is that there is a rotation by π/2 about the x-axis with a radial expansion by a factor of 2.

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A group of retired admirals, generals, and other senior military leaders, recently published a report, "Too Fat to Fight". The r

weqwewe [10]

Answer:

$z=\frac{0.694 -0.75}{\sqrt{\frac{0.75(1-0.75)}{180}}}=-1.735$

$p_v =P(z$

If we compare the p value obtained and the significance level given $\alpha=0.05$ we have $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of americans between 17 to 24 that not qualify for the military is significantly less than 0.75 or 75% .

Step-by-step explanation:

1) Data given and notation

n=180 represent the random sample taken

X=125 represent the number of americans between 17 to 24 that not qualify for the military

$\hat p=\frac{125}{180}=0.694$ estimated proportion of americans between 17 to 24 that not qualify for the military

$p_o=0.75$ is the value that we want to test

$\alpha=0.05$ represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

$p_v$ represent the p value (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that less than 75% of Americans between the ages of 17 to 24 do not qualify for the military :

Null hypothesis: $p\geq 0.75$

Alternative hypothesis: $p < 0.75$

When we conduct a proportion test we need to use the z statistic, and the is given by:

$z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}$ (1)

The One-Sample Proportion Test is used to assess whether a population proportion $\hat p$ is significantly different from a hypothesized value $p_o$ .

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

$z=\frac{0.694 -0.75}{\sqrt{\frac{0.75(1-0.75)}{180}}}=-1.735$

4) Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level provided $\alpha=0.05$ . The next step would be calculate the p value for this test.

Since is a left tailed test the p value would be:

$p_v =P(z$

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Answer:

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Step-by-step explanation:

The graph represents the function f(x) = x² + 3x + 2.

Now, g(x) is the function which is obtained by reflecting f(x) across the x-axis.

While a graph of a function reflects across x-axis then its y-values will change sign for a fixed value of x.

Therefore, the function g(x) will be given by

g(x) = - (x² + 3x + 2) = - x² - 3x - 2 (Answer)

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