given the linear differential system x ' = ax with determine if u, v form a fundamental solution set. if so, give the general solution to the system.

The required answer is the value of tan(2) is approximately -2352/3669.

To evaluate the expression under the given conditions, we will first determine the value of sin() using the Pythagorean identity and then use the double-angle formula for tan(2).
A Quadrant is circular sector of equal one quarter of a circle ,or  a half semicircle. A sector of two-dimensional cartesian  coordinate system.  The Pythagorean identity, are useful expression involving the function need to simplified.

Given: cos() = 7/25, and is in Quadrant I.
Step 1: Find sin()
Since we are in Quadrant I, sin() is positive. Using the Pythagorean identity, sin^2() + cos^2() = 1, we can find sin().
sin^2() + (7/25)^2 = 1
sin^2() = 1 - (49/625)
sin^2() = (576/625)
sin() = √(576/625) = 24/25

we  are called the Pythagorean identity is  Pythagorean trigonometric identity, is expression A to B .

The same value for all variables within certain range. Angle is double or multiply by 2 so we called double- angle.

Step 2: Find tan(2) using the double-angle formula
The double-angle formula for tangent is: tan(2) = (2 * tan()) / (1 - tan^2())
First, we find tan():
tan() = sin() / cos() = (24/25) / (7/25) = 24/7
Now, use the formula for tan(2):
tan(2) = (2 * (24/7)) / (1 - (24/7)^2)
tan(2) = (48/7) / (1 - 576/49)
tan(2) = (48/7) / ((49 - 576) / 49)
tan(2) = (48/7) * (49 / (-527))
tan(2) = (-2352 / 3669)
So, under the given conditions, the value of tan(2) is approximately -2352/3669.

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This is because the projection of a vector onto the Subspace it already belongs to is the vector itself. Therefore, Pw u = u.

To prove the statement, "for any u in Rn, Pw u = u if and only if u is in W," we need to demonstrate both directions of the "if and only if" statement.

Direction 1: If Pw u = u, then u is in W.

Assume that Pw u = u. We want to show that u is in W.

Recall that Pw u represents the projection of u onto the subspace W. If Pw u = u, it means that the projection of u onto W is equal to u itself.

By definition, if the projection of u onto W is equal to u, it implies that u is already in W. This is because the projection of u onto W gives the closest vector in W to u, and if the closest vector is u itself, then u must already be in W. Therefore, u is in W.

Direction 2: If u is in W, then Pw u = u.

Assume that u is in W. We want to show that Pw u = u.

Since u is in W, the projection of u onto W will be equal to u itself. This is because the projection of a vector onto the subspace it already belongs to is the vector itself. Therefore, Pw u = u.

By proving both directions, we have shown that "for any u in Rn, Pw u = u if and only if u is in W."

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We have proved both directions of the statement, and we can conclude that, for any u in Rn, Pw u = u if and only if u is in W.

To prove that, for any u in Rn, Pw u = u if and only if u is in W, we need to prove both directions of the statement.

First, let's assume that Pw u = u. We need to prove that u is in W. By definition, the projection of u onto W is the closest vector in W to u. If Pw u = u, then u is the closest vector in W to itself, which means that u is in W.

Second, let's assume that u is in W. We need to prove that Pw u = u. By definition, the projection of u onto W is the closest vector in W to u. Since u is already in W, it is the closest vector to itself, which means that Pw u = u.

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True. In a linear regression equation, when X is equal to the mean of X (MX), the predicted value of Y will be equal to the mean of Y (MY). This is because the regression equation aims to model the relationship between X and Y by finding the line that best fits the data points.

The regression equation takes the form of Y = bX + a, where b is the slope of the line and a is the y-intercept. When X is equal to its mean (MX), the term bX in the equation becomes b * MX, which simplifies to b * MX. Additionally, the y-intercept term a remains constant.

Since the mean of X (MX) is a fixed value, multiplying it by the slope (b) in the equation gives a constant term. This means that the predicted value of Y, when X is equal to its mean (MX), will be equal to a constant term plus the y-intercept (MY).

Therefore, when X = MX, the predicted value of Y in the regression equation is equal to MY, the mean of Y.

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Answer:

1.Neither

2.Perpendicular

3.Parallel

Step-by-step explanation:

y - 3 = 6(x + 2) Isn't anything,

y + 3 = -(1/3) Is definitely Perpendicular

(x - 4) Seems to be parallel.

This is one of my first times answering,I sure hope this helps!

The estimate of the area under the graph of f(x) = √(8x) using four rectangles is approximately [insert numerical value] square units.

To estimate the area under the graph of f(x) = √(8x) from x = 0 to x = 4 using four rectangles, we divide the interval [0, 4] into four equal subintervals: [0, 1], [1, 2], [2, 3], and [3, 4]. We then calculate the width of each rectangle by taking the difference between the x-coordinates of the endpoints of each subinterval, which is 1.

Next, we evaluate the function at the midpoint of each subinterval (0.5, 1.5, 2.5, and 3.5) to obtain the height of each rectangle. Taking the product of the width and height of each rectangle gives us the area of each rectangle. Finally, we sum up the areas of all four rectangles to get an estimate of the total area under the graph.

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The rate at which the tree will grow in just 1 year would be = 3ft/year.

The quantity of tree that grows in 1/12 year = 1/4 ft

The quantity of tree that will grow in 1 year = X ft.

That is;

1/12 years = 1/4ft

1 year = X

Make X the subject of formula;

X= 1/4÷1/12

X = 1/4×12/1

X = 3 ft

Therefore, the rate at which the tree will grow in just 1 year would be = 3 ft/year.

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The center of the circle is at (-3,4)

The radius is 6 which squared is 36.

So the equation is:

(x + 3)^2 + (y - 4)^2 = 36

[tex](x+3)^{2} +(y-4)^{2}=36[/tex]

The probability that an email tagged as spam is not actually spam is approximately 0.0481 or 4.81%.

To find the probability that an email tagged as spam is not actually spam, we can use conditional probability.

Let's define the following events:

A: Email is tagged as spam

B: Email is actually spam

We want to find P(not B | A), which represents the probability that an email is not spam (not B) given that it is tagged as spam (A).

Using Bayes' theorem, we have:

P(not B | A) = P(A | not B) * P(not B) / P(A)

P(A | not B) represents the probability that an email is tagged as spam (A) given that it is not actually spam (not B). From the problem statement, we know that this is 5% or 0.05.

P(not B) represents the probability that an email is not actually spam. Since 50% of emails are spam, the probability that an email is not spam is 1 - P(B) = 1 - 0.50 = 0.50.

P(A) represents the probability that an email is tagged as spam. This can be calculated by considering two cases: an email is spam and correctly tagged as spam, or an email is not spam and incorrectly tagged as spam. Mathematically, we can write:

P(A) = P(A | B) * P(B) + P(A | not B) * P(not B)

P(A | B) represents the probability that an email is tagged as spam (A) given that it is actually spam (B). From the problem statement, we know that this is 99% or 0.99.

P(B) represents the probability that an email is actually spam. From the problem statement, we know that this is 50% or 0.50.

Plugging in the values, we can calculate P(A):

P(A) = P(A | B) * P(B) + P(A | not B) * P(not B)

= 0.99 * 0.50 + 0.05 * 0.50

= 0.495 + 0.025

= 0.52

Now, we can calculate P(not B | A):

P(not B | A) = P(A | not B) * P(not B) / P(A)

= 0.05 * 0.50 / 0.52

= 0.025 / 0.52

≈ 0.0481

Therefore, the probability that an email tagged as spam is not actually spam is approximately 0.0481 or 4.81%.

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a) p^ = 27/75 = 0.36

Including the sampling of the (b) parameter in the Gibbs sampler can lead to issues with convergence.

In the Gibbs sampler, as it progresses, the samples are assumed to be drawn from the prior conditional distributions given the observed data. However, if the sampling of a particular variable is included, such as the (b) parameter, the Gibbs sampler may not converge.

The Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm used for drawing samples from a joint distribution when the conditional distributions are easier to sample from individually. In each iteration of the Gibbs sampler, the values of variables are updated one at a time based on their conditional distributions.

Ideally, the Gibbs sampler aims to converge to the target distribution, allowing for efficient estimation and inference. However, the inclusion of certain variables in the sampling process can affect the convergence properties of the sampler. Specifically, if the (b) parameter is sampled in the Gibbs sampler, it may prevent convergence.

The convergence of the Gibbs sampler relies on the Markov chain satisfying certain conditions, such as irreducibility, aperiodicity, and ergodicity. When a parameter like (b) is included, it may introduce dependencies or correlations that violate these conditions, preventing the sampler from reaching a stationary distribution.

Therefore, including the sampling of the (b) parameter in the Gibbs sampler can lead to issues with convergence. It is important to carefully consider the impact of including or excluding variables in the sampling process and assess the convergence properties of the Gibbs sampler in each specific case.

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The triangle that is similar to triangle ABC is triangle DEF.

Similar triangles are defined as the triangles that have the same shape, but their sizes may vary.

This means that all equilateral triangles, squares of any side lengths are examples of similar objects.

Therefore, we can say that if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

We want to find the triangle that ois similar to triangle ABC.We see that:

∠A = 37°

∠B = 94°

From the options, we see in the first option that

∠D = 37°

∠E = 94°

Thus, triangle DEF is similar to Triangle ABC.

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Based on the equation of the curve of best fit above, the amount of overall points LeBron James would have at the end of his career is 28,062 points.

In this exercise, we would plot the rookie season-points on the x-axis (x-coordinates) of a scatter plot while the overall points would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

On the Microsoft Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation of the curve of best fit (trend line) on the scatter plot.

Based on the scatter plot shown below, which models the relationship between the rookie season-points and the overall points, an equation of the curve of best fit is modeled as follows:

y = 5.74x + 18568

Based on the equation of the curve of best fit above, the amount of overall points LeBron James would have at the end of his career can be calculated as follows;

y = 5.74x + 18568

y = 5.74(1,654) + 18568

y = 28,061.96 ≈ 28,062 points.

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To start, let's sketch the graph of the curve y = x from x = 0 to x = 20. This is simply a diagonal line that passes through the points (0,0) and (20,20), as shown below:

```
   |
20  |    *
   |   *  
   |  *  
   | *    
   |*    
0  --------------
  0   10   20
```

Now, we want to revolve this curve around the x-axis to create a solid shape. Specifically, we want to create a paraboloid, which is a three-dimensional shape that looks like an upside-down bowl.

To find the volume of this paraboloid, we need to use calculus. The basic idea is to slice the solid into very thin disks, and then add up the volumes of all the disks to get the total volume.

To do this, we'll use the formula for the volume of a cylinder, which is:

V = πr^2h

where r is the radius of the cylinder and h is its height. In our case, each disk is a cylinder with radius r and height h, where:

- r is equal to the y-value of the curve (i.e. r = y = x), since the disk extends from the x-axis to the curve.
- h is the thickness of the disk, which is a very small change in x. We can call this dx.

So, the volume of each disk is:

dV = πr^2dx
  = πx^2dx

To find the total volume of the paraboloid, we need to add up the volumes of all the disks. This is done using an integral:

V = ∫(from x=0 to x=20) dV
 = ∫(from x=0 to x=20) πx^2dx

Evaluating this integral gives us:

V = π/3 * 20^3
 = 8000π/3

So the exact volume of the paraboloid is 8000π/3.

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To find the average cost per item to produce 50 items, we need to divide the total cost of manufacturing 50 items by 50.

First, let's plug in x = 50 into the cost function C = 100 + 80x:

C = 100 + 80(50)

C = 100 + 4000

C = 4100

So, it costs $4100 to manufacture 50 items.

To find the average cost per item, we divide the total cost by the number of items:

Average cost per item = total cost / number of items

Average cost per item = $4100 / 50

Average cost per item = $82

Therefore, the average cost per item to produce 50 items is $82.

It's worth noting that the cost function given in the question assumes that the cost of manufacturing each item remains constant as more items are produced. This is known as the "constant marginal cost assumption". In reality, however, the cost to manufacture each additional item may increase due to factors such as diminishing returns or economies of scale.

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The maximum annual profit is 1,72,0000

The profit function can be found by subtracting the cost function from the revenue function:

[tex]P(x) = R(x) - C(x) = (520x - 0.02x^2) - (160x + 100,000) = -0.02x^2 + 360x - 100,000[/tex]

To find the maximum annual profit, we need to find the value of x that maximizes the profit function.

One way to do this is to find the vertex of the parabola given by the profit function.

The x-coordinate of the vertex is given by:

x = -b/2a

where a = -0.02 and b = 360.

Substituting these values, we get:

[tex]x = -360/(2\times (-0.02)) = 9,000[/tex].

Therefore, the manufacturer should make 9,000 drives to maximize annual profit.

To find the maximum profit,  we can substitute this value into the profit function:

[tex]P(9,000) = -0.02(9,000)^2 + 360(9,000) - 100,000 = $1,720,000[/tex]

Therefore, the answer is (B) $1,720,000.

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To compute (p,q) with the given inner product, we need to evaluate p(1), p(0), p(-1), q(1), q(0), and q(-1), and use them to form the dot product of the coordinate vectors [p(1), p(0), p(-1)] and [q(1), q(0), q(-1)].

Using p(t) = 6-t, we get p(1) = 5, p(0) = 6, and p(-1) = 7. Using q(t) = 3 + 2t^2, we get q(1) = 5, q(0) = 3, and q(-1) = 5. Therefore, the coordinate vectors are [5, 6, 7] and [5, 3, 5], and their dot product is (5)(5) + (6)(3) + (7)(5) = 80. Thus, (p,q) = 80.

In general, an inner product on a vector space V is a function that takes two vectors v and w in V and returns a scalar (v,w) satisfying certain properties, such as linearity in the first argument, symmetry, and positive-definiteness. One common example of an inner product on the vector space of polynomials of degree at most n is the evaluation inner product, which is defined as (p,q) = ∫[a,b] p(x)q(x) dx, where [a,b] is some interval and the integral is taken over that interval. However, if we restrict our attention to the subspace of polynomials of degree at most 2, we can define a simpler inner product by evaluating the polynomials at certain points and taking the dot product of the resulting coordinate vectors. This inner product has the advantage of being easy to compute and visualize.

To compute the inner product of two polynomials p and q with the given inner product, we evaluate the polynomials at the points 1, 0, and -1, and use the resulting coordinates to form the dot product. This yields a scalar that represents the angle between the two polynomials in a sense. In this case, we found that the inner product of p(t) = 6-t and q(t) = 3 + 2t^2 is (p,q) = 80. This means that the angle between p and q is relatively small, since the dot product is positive and relatively large. However, the precise meaning of this angle is not immediately clear without further context or geometric interpretation.

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A standard normal distribution table or calculator, the p-value for z = 5.98 is less than 0.0001, which is much smaller than the typical alpha level of 0.05. Option (d) is the correct answer.

To determine if the proportion of 12th-graders who eat breakfast daily is lower than the proportion of 9th-graders, we need to conduct a hypothesis test. Let p1 and p2 be the true population proportions of 9th and 12th graders who eat breakfast daily, respectively. Our null hypothesis is that the two population proportions are equal, i.e. H0: p1 = p2, and the alternative hypothesis is that the proportion of 12th graders is lower, i.e. Ha: p1 < p2.

We can use a z-test to compare the proportions. The test statistic is given by

z = (p1 - p2) / sqrt(p_hat * (1 - p_hat) * (1/n1 + 1/n2))

where p_hat = (x1 + x2) / (n1 + n2), x1 and x2 are the number of 9th and 12th graders who ate breakfast daily, respectively, and n1 and n2 are the sample sizes.

Plugging in the values given in the problem, we get:

p1 = 1374/3470 = 0.396

p2 = 1116/3301 = 0.338

n1 = 3470, n2 = 3301

p_hat = (1374 + 1116) / (3470 + 3301) = 0.367

z = (0.396 - 0.338) / sqrt(0.367 * (1 - 0.367) * (1/3470 + 1/3301)) = 5.98

Using a standard normal distribution table or calculator, the p-value for z = 5.98 is less than 0.0001, which is much smaller than the typical alpha level of 0.05.

Therefore, we reject the null hypothesis and conclude that there is evidence to suggest that the proportion of 12th-graders who eat breakfast daily is lower than the proportion of 9th-graders eating breakfast daily. The numerical value of the z statistic for comparing the proportions of 9th- and 12th-graders who ate breakfast daily is 5.98.  Option (d) is the correct answer.

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The numerical value of the z statistic for comparing the proportions of 9th- and 12th-graders who ate breakfast daily is 3.86.

To test whether the proportion of 12th-graders who eat breakfast daily is lower than the proportion of 9th-graders, we can use a hypothesis test with the following null and alternative hypotheses:

H0: P9 = P12

Ha: P9 < P12

where P9 and P12 are the true proportions of all 9th- and 12th-graders who eat breakfast daily.

We can use a z-test for the difference between two proportions to test this hypothesis. The formula for the test statistic is:

z = (p1 - p2) / SE

where p1 and p2 are the sample proportions, and SE is the standard error of the difference between the proportions:

SE = sqrt(p(1-p) / n1 + p(1-p) / n2)

where p is the pooled proportion of successes, defined as:

p = (x1 + x2) / (n1 + n2)

and x1, x2, n1, and n2 are the number of successes and sample sizes for the two groups.

Plugging in the values from the problem, we have:

p1 = 1374 / 3470 = 0.396

p2 = 1116 / 3301 = 0.338

n1 = 3470

n2 = 3301

p = (1374 + 1116) / (3470 + 3301) = 0.368

SE = sqrt(0.368(1-0.368) / 3470 + 0.368(1-0.368) / 3301) = 0.015

z = (0.396 - 0.338) / 0.015 = 3.86

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Answer:

a) (0,1)

[tex]\sf b) k = \dfrac{13}{2}[/tex]

Step-by-step explanation:

a) The x co-ordinate where the line (y -4x = 1) crosses the y-axis is zero.

       y - 4*0 = 1

             y = 1

co-ordinates (0,1)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

b) y = 2x + c

Compare with y = mx + c.

⇒ m = 2

Two points from the graph: (k , 10) & (0,-3)

Substitute the value of m and the two points in the below formulae and find the value of k.

        [tex]\sf slope =\dfrac{y_2 -y_1}{x_2-x_1}[/tex]

       [tex]\dfrac{-3-10}{0-k}=2\\\\\dfrac{-13}{-k}=2\\\\\\\dfrac{13}{k}=2\\\\\\Cross \ multiply,\\\\[/tex]

              13 = 2k

              [tex]\sf\boxed{ \bf k =\dfrac{13}{2}}\\\\[/tex]

A. P(6, then 1) = 1/90

B. P(even, then 5) = 1/18

C. P(8, then odd) = 1/18

D. P(3, then prime) = 2/45

E. P(prime, composite) = 4/15

F. P(even, then 3, then 5) = 1/144

Given:

Total number of cards: 10

A. P(6, then 1):

P(6, then 1) = 1/10 x 1/9

= 1/90

B. P(even, then 5):

Number of favorable outcomes: 5 x 1 = 5

P(even, then 5) = 5/10 x 1/9

= 1/18

C. P(8, then odd):

Number of favorable outcomes: 1 x 5 = 5

P(8, then odd) = 1/10 x 5/9

= 1/18

D. P(3, then prime):

Number of favorable outcomes: 1 x 4 = 4

P(3, then prime) = 1/10 x 4/9

= 2/45

E. P(prime, composite):

Number of favorable outcomes: 4 x 6 = 24

P(prime, composite) = 4/10 x 6/9

= 4/15

F. P(even, then 3, then 5):

Number of favorable outcomes: 5 x 1 x 1 = 5

P(even, then 3, then 5) = 5/10 x 1/9 x 1/8

= 1/144

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You can make a maximum of two tomato-cheese sandwiches. because you can only make as many tomato-cheese sandwiches as the number of cheese slices you have

To make a tomato-cheese sandwich, you need one tomato slice and one cheese slice. Since you have three tomato slices and two cheese slices, you are limited by the availability of cheese slices.

Therefore, you can only make as many tomato-cheese sandwiches as the number of cheese slices you have, which in this case is two.

The ingredient that limits the number of sandwiches you can make is the cheese slice. You have more tomato slices than cheese slices, so you cannot make more than two tomato-cheese sandwiches.

Even if you have extra tomato slices, you cannot make additional sandwiches because you do not have enough cheese slices to pair with them.

In summary, the number of tomato-cheese sandwiches you can make is determined by the ingredient with the lowest quantity,

which in this case is the cheese slice. Therefore, you can make a maximum of two tomato-cheese sandwiches.

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The X value corresponding to z = -2.00 is 60. The X value corresponding to z = -2.00 is 60. This means that the observation with a z-score of -2.00 is 60 units below the population mean of 80.

To find the X value corresponding to z = -2.00, we can use the formula:
z = (X - µ) / σ
Substituting the given values, we get:
-2.00 = (X - 80) / 10
Solving for X, we get:
X = (-2.00 x 10) + 80
X = 60

The z-score measures the number of standard deviations an observation is from the mean. In this case, the given z-score of -2.00 indicates that the observation is 2 standard deviations below the mean.
To find the corresponding X value, we use the formula:
z = (X - µ) / σ
Where z is the standard normal distribution value, X is the corresponding raw score, µ is the mean of the population, and σ is the standard deviation of the population.
Substituting the given values, we get:
-2.00 = (X - 80) / 10
Solving for X, we get:
X = (-2.00 x 10) + 80
X = 60

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To serve eight people, you would need 1/2 cup of grated parmesan cheese.

To expand the recipe to feed eight people, we need to determine the amount of parmesan cheese needed for the new serving size.

Given that the original recipe for lasagna serving four requires 1/4 cup of grated parmesan cheese, we can calculate the amount needed to serve eight people by scaling up the recipe.

If the original recipe serves four and requires 1/4 cup of grated parmesan cheese, then for eight servings, we would need to double the recipe.

Doubling the recipe means doubling all the ingredients, including the parmesan cheese.

1/4 cup * 2 = 1/2 cup

Therefore, to serve eight people, you would need 1/2 cup of grated parmesan cheese.

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Answer:

1) x= 4, -1

2) x= 1, -2

3) x= 2,1,-1

4) x= -3,1

Answer:     M=3

Step-by-step explanation:

The slope-intercept form is y=mx+b, where mis the slope and b is the y-intercept. y=mx+b

Simplify the right side.

That sentence translates to 3+4x where x is the unknown number.

The term "sum" refers to "the result of adding". The 4x means 4*x or "4 times x". Other letters can be used as the variable.

1000 cubic centimeters would need to be placed in a tub of water to displace 1 Lter of water

Conversion of units simply refers to the method used in determining the equivalent of one unit in relation to another.

From the information given, we have that;

Number of cubic centimeters that would be placed in a tub of water to displace 1 L of water

So, we have that there is 1 liter of water in the tub

In order to displace, you need to put something in that is the same amount

Now, let's convert the units

1 liter = 1000 cubic cm

Hence, you need 1000 cubic cm to displace 1 liter

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The probability that a patient, randomly selected and not allergic to pollen, tested negative is 0.90.

To find the probability that a patient tested negative given that they are not allergic to pollen, we can use Bayes' theorem:

P(N|A complement) = [P(N complement|A complement) × P(A complement)] / P(N complement)

We know that P(A complement) = 1 - P(A) = 1 - 0.10 = 0.90. Additionally, P(N complement) can be calculated as:

P(N complement) = P(N complement|A) × P(A) + P(N complement|A complement) × P(A complement)

= 0.10 × 0.10 + 0.20 × 0.90

= 0.01 + 0.18

= 0.19

Substituting these values into the formula, we have:

P(N|A complement) = (0.20 × 0.90) / 0.19 = 0.90

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Matrix A = xd[tex]x^{-1}[/tex] is [tex]\left[\begin{array}{cc}4/\sqrt{17} &2/\sqrt{5} \\1/\sqrt{17} &1/\sqrt{5} \end{array}\right][/tex] [tex]\left[\begin{array}{cc}0 &0 \\0 &-2 \end{array}\right][/tex] [tex]\left[\begin{array}{cc}1/\sqrt{17} &-2/\sqrt{85} \\-1/\sqrt{17} &4/\sqrt{85} \end{array}\right][/tex] .

For the matrix A =

[ 2 -8 ]

[ 1 -4 ]

we need to find x and d such that A = xd[tex]x^{-1}[/tex].

First, we find the eigenvalues of A:

det(A - λI) = (2 - λ)(-4 - λ) - (-8)(1) = λ*λ + 2λ = λ(λ + 2) = 0

So, the eigenvalues are λ1 = 0 and λ2 = -2.

Next, we find the eigenvectors associated with each eigenvalue:

For λ1 = 0:

(A - λ1I)x = 0

[ 2 -8 ] [x1] [0]

[ 1 -4 ] [x2] = [0]

Solving for x gives x = [tex][4,1]^{T}[/tex].

For λ2 = -2:

(A - λ2I)x = 0

[ 4 -8 ] [x1] [0]

[ 1 -3 ] [x2] = [0]

Solving for x gives x = [tex][2,1]^{T}[/tex].

We normalize the eigenvectors to get x1 = [tex][4/\sqrt{17},1/\sqrt{17} ]^{T}[/tex] and x2 = [tex][2/\sqrt{5},1/\sqrt{5} ]^{T}[/tex] .

Now, we can find d:

d = [λ1 0; 0 λ2] = [0 0; 0 -2]

Finally, we can find [tex]x^{-1}[/tex]:

[tex]x^{-1}[/tex]  = [tex]\left[\begin{array}{cc}4/\sqrt{17} &2/\sqrt{5} \\1/\sqrt{17} &1/\sqrt{5} \end{array}\right]^{-1}[/tex] =  [tex]\left[\begin{array}{cc}1/\sqrt{17} &-2/\sqrt{85} \\-1/\sqrt{17} &4/\sqrt{85} \end{array}\right][/tex]

Therefore, we have:

A = xd[tex]x^{-1}[/tex]  = [tex]\left[\begin{array}{cc}4/\sqrt{17} &2/\sqrt{5} \\1/\sqrt{17} &1/\sqrt{5} \end{array}\right][/tex] [tex]\left[\begin{array}{cc}0 &0 \\0 &-2 \end{array}\right][/tex] [tex]\left[\begin{array}{cc}1/\sqrt{17} &-2/\sqrt{85} \\-1/\sqrt{17} &4/\sqrt{85} \end{array}\right][/tex]

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