suppose that AB is invertible then (AB)^−1 exists. We also know (AB)^−1=B^−1A^−1. If we let C=(B^−1A−^1A) then by the invertible matrix theorem we see that since CA=I(left inverse) then B is invertible. Would this be correct?

Hope, Casey, and Daniela are the players in order from lowest to highest points per game average.

To find the points per game for each player, we can divide their total points by the number of games they played:

Casey: 154 points ÷ 8 games = 19.25 points per game

Hope: 132 points ÷ 7 games = 18.86 points per game

Daniela: 101 points ÷ 5 games = 20.2 points per game

Casey mistakenly ordered the players as follows: Hope, Casey, and then Hope. This ordering is incorrect because Casey had a higher point-per-game average than Hope.

The correct ordering from least to greatest points per game average is: Hope, Casey, and then Daniela.

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Complete question:

Daniela scored

101 points in 5 basketball games. Casey scored 154 points in 8 games. Hope scored 132 points in 7 games. Casey tried to order the players by their points per game from least to greatest, but he made a mistake. Here's his work:

The Maclaurin series for the given function can be obtained using the formula for the Maclaurin series and the expression given in the table.

To obtain the Maclaurin series for the given function, we need to use the formula for the Maclaurin series, which is:

f(x) = ∑ (n=0 to infinity) [ f^(n)(0) / n! ] * x^n

where f^(n)(0) is the nth derivative of f(x) evaluated at x = 0.

Using the formula given in the table, we have:

f(x) = ∑ (n=0 to infinity) [ (1! * 2! * 3! * ... * (2n-1)) / ((2n)! * (2n+1)) ] * x^(2n+1)

Simplifying the expression, we have:

f(x) = ∑ (n=0 to infinity) [ (-1)^n * (2n)! / (2^(2n+1) * (n!)^2 * (2n+1)) ] * x^(2n+1)

Therefore, the Maclaurin series for the given function is:

f(x) = x - (1/3)x^3 + (1/5)x^5 - (1/7)x^7 + ...

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Answer:

Step-by-step explanation:

Answer:

5%

Step-by-step explanation:

he earned 50 so his investment grew from 1000 to 1050

now simply calculate the percentage change

(1050-1000)/1000 *100

=5%

The pH during the titration of 20.44 mL of 0.26 M HBr with 0.14 M KOH after 10.97 mL of the base have been added is approximately 0.4001.

To calculate the pH during the titration of 20.44 mL of 0.26 M HBr with 0.14 M KOH after 10.97 mL of the base have been added, we need to use the following equation:

n(HBr) x V(HBr) x M(HBr) = n(KOH) x V(KOH) x M(KOH)

where n is the number of moles, V is the volume, and M is the molarity.

First, we need to calculate the number of moles of HBr in the initial solution:

n(HBr) = M(HBr) x V(HBr)
n(HBr) = 0.26 mol/L x 0.02044 L
n(HBr) = 0.0053144 mol

Next, we need to calculate the number of moles of KOH added:

n(KOH) = M(KOH) x V(KOH)
n(KOH) = 0.14 mol/L x 0.01097 L
n(KOH) = 0.0015358 mol

Since KOH is a strong base and HBr is a strong acid, they will react in a 1:1 ratio, so the number of moles of HBr that remain after the addition of KOH will be:

n(HBr) remaining = n(HBr) - n(KOH)
n(HBr) remaining = 0.0053144 mol - 0.0015358 mol
n(HBr) remaining = 0.0037786 mol

Now we can calculate the volume of the remaining HBr solution:

V(HBr) remaining = V(HBr) - V(KOH)
V(HBr) remaining = 0.02044 L - 0.01097 L
V(HBr) remaining = 0.00947 L

Finally, we can calculate the new concentration of the HBr solution:

M(HBr) = n(HBr) remaining / V(HBr) remaining
M(HBr) = 0.0037786 mol / 0.00947 L
M(HBr) = 0.3988 M

To calculate the pH, we need to use the following equation:

pH = -log[H+]

where [H+] is the concentration of hydrogen ions.

Since HBr is a strong acid, it dissociates completely in water to form H+ and Br- ions, so the concentration of H+ ions is equal to the concentration of the remaining HBr solution:

[H+] = M(HBr)
[H+] = 0.3988 M

pH = -log(0.3988)
pH = 0.4001

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The dimensions of the billboard are 3 feet by 0.875 feet. The area of the billboard is 2.625 square feet.

To find the dimensions of the billboard in feet, we need to scale down the width and length of the advertisement by a factor of 4.

Width of the billboard in feet = (144 inches / 4) / 12 inches/foot = 3 feet

Length of the billboard in feet = (42 inches / 4) / 12 inches/foot = 0.875 feet

Therefore, the dimensions of the billboard are 3 feet by 0.875 feet.

To find the area of the billboard in square feet, we multiply the width and length of the billboard in feet.

Area of the billboard = width x length = 3 feet x 0.875 feet = 2.625 square feet

Therefore, the area of the billboard is 2.625 square feet.

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Answer:

Part A:

144 x 4 = 576 12 = 48 ft 

so 48 ft is the width 

42 x 4 = 168 12 = 14 ft

so the length is 14 ft

48ft by 14ft

Part B:

48 x 14 = 672 ft2 

Step-by-step explanation:

Answer: 3055

Step-by-step explanation:

47 x 65

Separate the 40 and the 7, and the 60 and the 5. You get 40 from the 4 in 47 because 4 is in the tens place, you get 60 from 65 for the same reason. Then you want to multiply 40 x 60, which can also be 6 x 4, then add two 0's. So for this we have 2400. Then multiply the 40 by the 5, this can be 4 x 5 then add a 0. Now for this we have 200. Now you want to multiply the 7 by the 60. 7 x 6 = 42, so add a 0.

7 x 60 = 420

Now, multiply the 7 by the 5, this is 35. Lastly, you want to add all the products together.

35 + 420 + 200 + 2400

= 3055

The apothem of a regular pentagon with a side length of 6 is approximately 4.37614 units.

Given the side length of a regular hexagon, we may use one of these formulas to get its apothem. Consider a normal hexagon with 7 inches of side length as an example.

We can use the following formula to determine the apothem of a regular pentagon with six sides:

apothem=(side length)/(2*tan(pi/number of sides))

Five sides make up a normal pentagon, and pi is about 3.14159. When these values and the 6 side length are entered into the formula, we obtain:

apothem = (6) / (2 * tan(pi / 5))

apothem = (6) / (2 * tan(3.14159 / 5))

apothem = (6) / (2 * 0.68819)

apothem = 4.37614

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Using a t-distribution table, here with 46 degrees of freedom (n-1), the critical value for a two-tailed test with a level of significance of α = 0.10 is ±2.575. Therefore, we reject H0 if the standardized test statistic is greater than 2.575 or less than -2.575. The correct answer is D. Reject H0 if the standardized test statistic is greater than 2.575 or less than -2.575.

To determine whether to reject H0, we need to calculate the standardized test statistic using the formula (sample mean - hypothesized mean) / (standard deviation / square root of sample size). Since the null hypothesis is that μ = 3.5, the sample mean and standard deviation must be used to calculate the standardized test statistic.
Assuming a normal distribution, with a sample size of n=47 and a level of significance of α = 0.10, we can use a two-tailed test with a critical value of ±1.645. However, since we are testing for μ ≠ 3.5, this is a two-tailed test, and we need to use a critical value that accounts for both tails.

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Answer: 2,620 meatballs.

Step-by-step explanation:

Initially, Harold had 150 meatballs. Sarah ate 30 of them, so there are 150 - 30 = 120 meatballs left. The waiter then brought 2,500 more meatballs. Therefore, the total number of meatballs now is 120 + 2,500 = 2,620 meatballs.

The given sequence alpha_n = ln(9n/n + 1) converges, and its limit is 1.

To determine if the sequence alpha_n = ln(9n/n + 1) converges or diverges, we can find the limit as n approaches infinity.

Step 1: Rewrite the expression using properties of logarithms:
alpha_n = ln(9n) - ln(n + 1)

Step 2: As n approaches infinity, both terms will also approach infinity, but we can analyze their behavior by finding the limit of their ratio:

Lim (n -> infinity) (ln(9n) / ln(n + 1))

Step 3: Apply L'Hopital's Rule since it's an indeterminate form:

Lim (n -> infinity) (d/dn ln(9n) / d/dn ln(n + 1))

Step 4: Calculate the derivatives of the numerator and denominator:

d/dn ln(9n) = (9 / (9n))
d/dn ln(n + 1) = (1 / (n + 1))

Step 5: Find the limit:

Lim (n -> infinity) ((9 / (9n)) / (1 / (n + 1)))

Step 6: Simplify the limit expression:

Lim (n -> infinity) ((9 / 9n) * (n + 1))

Step 7: Simplify further and take the limit:

Lim (n -> infinity) (n + 1) / n = 1

So, the sequence alpha_n = ln(9n/n + 1) converges, and its limit is 1.

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Answer:

98.59006

Step-by-step explanation:

simplify to what it asks, since you didn't give us what to simplify to

A Type II error in this setting is the mean amount of [tex]CO_2[/tex] emitted by the new fuel is actually lower than 89 kg but they fail to conclude it is lower than 8.9 kg. Option B is right choice.

In hypothesis testing, a Type II error occurs when we fail to reject a false null hypothesis. In this case, the null hypothesis is that the mean amount of  [tex]CO_2[/tex] emitted by the new gasoline is 8.9 kg, while the alternative hypothesis is that the mean is less than 8.9 kg.

Therefore, a Type II error would occur if the mean amount of [tex]CO_2[/tex] emitted by the new fuel is actually lower than 8.9 kg, but the test fails to reject the null hypothesis that the mean is 8.9 kg.

This means that the test fails to detect the difference in CO2 emissions between the new fuel and the standard fuel, even though the new fuel has lower  [tex]CO_2[/tex] emissions.

Option B is the correct answer because it describes this scenario - the mean amount of  [tex]CO_2[/tex] emitted by the new fuel is actually lower than 8.9 kg but they fail to conclude it is lower than 8.9 kg. This is a Type II error because the test fails to detect a true difference between the mean    [tex]CO_2[/tex] emissions of the new fuel and the standard fuel.

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The missing option are

a.The mean amount of CO2 emitted by the new fuel is actually 8.9 kg but they conclude it is lower than 8 9 kg

b. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fail to conclude it is lower than 8.9 kg

c. The mean amount of CO2 emitted by the new fuel is actually 8.9 kg and they fail to conclude it is lower than 8.9 kg

d. The mean amount of CO2 emitted by the new fuel is actually lower than 8 9 kg and they conclude it is lower than 8 9 kg

The maximum speed of the object at x = 6a is 3.12 m/s.

To find the maximum speed, we need to consider the conservation of mechanical energy. At x = 6a, the object's potential energy (PE) is given by the graph. Let's assume the difference in potential energy between 4a and 6a is ΔPE.

1. Calculate ΔPE: 4.91 J is the energy difference between adjacent grid lines, and the object moves two grid lines (from 4a to 6a). So, ΔPE = 4.91 J * 2 = 9.82 J.

2. Determine the object's kinetic energy (KE) at x = 6a: Since the mechanical energy is conserved, the increase in PE will be equal to the decrease in KE. Thus, KE = ΔPE = 9.82 J.

3. Calculate the maximum speed: Using the formula KE = 0.5 * mass * speed^2, we can find the maximum speed: 9.82 J = 0.5 * 0.42 kg * speed^2. Solving for speed, we get 3.12 m/s.

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We have shown that the null geodesics in a 3-space (x*)-(x, y, z) and line element with coordinates are given by: [tex]ds^2 = I(dx)^2 + 2ldxdy + 2mdxdz + 2m'dydz + ndy^2 + n'dz^2[/tex]= 0. where I, l', m, m', n, n' are arbitrary constants satisfying [tex]12m^2 < 2I[/tex]n.

Null geodesics in a 3-space (x*)-(x, y, z) and line element with coordinates are given by:

[tex]ds^2 = I(dx)^2 + 2ldxdy + 2mdxdz + 2m'dydz + ndy^2 + n'dz^2 = 0[/tex]

where I, l', m, m', n, n' are arbitrary constants satisfying[tex]12m^2 < 2In[/tex].

To prove this, we need to show that any solution to the above equation is a null geodesic, and any null geodesic can be expressed in this form.

Let's first assume that we have a solution to the above equation. We can write it in terms of a parameter u, such that:

x = x0 + Au
y = y0 + Bu
z = z0 + Cu

where A, B, and C are constants. Substituting these expressions into the line element equation, we get:

[tex]I(A^2)u^2 + 2lAuB + 2mAuC + 2m'BuC + n(B^2)u^2 + n'(C^2)u^2 = 0[/tex]

Since this equation holds for any value of u, each term must be zero. Therefore, we get six equations:

[tex]I(A^2) + 2lAB + 2mAC = 0[/tex]

[/tex]2m'BC + n(B^2) = 0[/tex]

[/tex]n'(C^2) = 0[/tex]

From the last equation, we get either C = 0 or n' = 0. If n' = 0, then the equation reduces to:

[/tex]I(A^2) + 2lAB + 2mAC + n(B^2) = 0[/tex]

which is the equation of a null geodesic in this 3-space. If C = 0, then we can write the line element equation as:

[/tex]I(dx)^2 + 2ldxdy + 2m'dydz + ndy^2 = 0[/tex]

which is also the equation of a null geodesic.

Now, let's assume we have a null geodesic in this 3-space. We can write it in the form:

x = x0 + Au
y = y0 + Bu
z = z0 + Cu

where A, B, and C are constants. Substituting these expressions into the line element equation, we get:

[/tex]I(A^2) + 2lAB + 2mAC + n(B^2) + 2m'BC + n'(C^2) = 0[/tex]

Since the geodesic is null, ds^2 = 0. Therefore, we get:

[/tex]ds^2 = I(dx)^2 + 2ldxdy + 2mdxdz + 2m'dydz + ndy^2 + n'dz^2 = 0[/tex]

which is the same as the line element equation we started with. Therefore, any null geodesic in this 3-space can be expressed in the form given by the equation above.

In conclusion, we have shown that the null geodesics in a 3-space (x*)-(x, y, z) and line element with coordinates are given by:

[/tex]ds^2 = I(dx)^2 + 2ldxdy + 2mdxdz + 2m'dydz + ndy^2 + n'dz^2 = 0[/tex]

where I, l', m, m', n, n' are arbitrary constants satisfying [/tex]12m^2 < 2In.[/tex]

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The correct flowchart with missing statements and reasons proves that the measure of angle ECB is 22°:

C.

1. Alternate interior angles are congruent

2. Triangle Sum Theorem

3. Measure of angle AED is 22 degrees

Alternate interior angles are congruent - This is correct as angle AED and angle ECB are alternate interior angles formed by a transversal (segment DE) intersecting two parallel lines (segment BC and segment AE), and thus they are congruent.

Since angle ADE is given to be 68 degrees, by substituting the value of angle AED (which is congruent to angle ECB) into the statement, we can conclude that the measure of angle ECB is 22 degrees using the Substitution Property.

Measure of angle AED is 22 degrees - This is correct as given in the problem statement.

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See correct option below

Triangle ABC is a right triangle. Point D is the midpoint of side AB, and point E is the midpoint of side AC. The measure of angle ADE is 68°.

Triangle ABC with segment DE. Angle ADE measures 68 degrees.

The following flowchart with missing statements and reasons proves that the measure of angle ECB is 22°:

Statement, Measure of angle ADE is 68 degrees, Reason, Given, and Statement, Measure of angle DAE is 90 degrees, Reason, Definition of right angle, leading to Statement 3 and Reason 2, which further leads to Statement, Measure of angle ECB is 22 degrees, Reason, Substitution Property. Statement, Segment DE joins the midpoints of segment AB and AC, Reason, Given, leading to Statement, Segment DE is parallel to segment BC, Reason, Midsegment theorem, which leads to Angle ECB is congruent to angle AED, Reason 1, which further leads to Statement, Measure of angle ECB is 22 degrees, Reason, Substitution Property.

Which statement and reason can be used to fill in the numbered blank spaces?

A.

1. Corresponding angles are congruent

2. Triangle Sum Theorem

3. Measure of angle AED is 22 degrees

B.

1. Corresponding angles are congruent

2. Base Angle Theorem

3. Measure of angle AED is 68 degrees

C.

1. Alternate interior angles are congruent

2. Triangle Sum Theorem

3. Measure of angle AED is 22 degrees

D.

1. Alternate interior angles are congruent

2. Triangle Angle Sum Theorem

3. Measure of angle AED is 68 degrees

The probability that Devin, Erin, and Hana will be selected again is 1 / 10.

An adept approach to calculating the chance of Devin, Erin, and Hana being picked again is through the utilization of the combinations formula. This efficient formula calculates the total amount of possible ways 3 pupils can be selected from a group of 5:

C ( n, k ) = n ! / (k ! ( n - k ) ! )

C (5, 3) = 5 ! / (3 !( 5 - 3 ) ! )

= 120 / 12

= 10

The probability that Devin, Erin, and Hana will be selected again is:

=  Number of ways they can be selected / Total number of ways to select 3 students

= 1 / 10

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We have proven that ∠JKM = ∠NKL.

To prove that ∠JKM = ∠NKL,

       J

      / \

     /   \

    /     \

   K-------N

  / \       / \

 /    \     /     \

/      \   /         \

M-------K-------L

To prove that JKM is congruent to NKL, we need to show that all corresponding sides and angles are equal.

First, we know that KM bisects JKN, so angle JKM is congruent to angle NKM (by the angle bisector theorem). Similarly, KN bisects MKL, so angle LKN is congruent to angle MKN.

Also, we know that JK is equal to KN (by the definition of a bisector), and KM is equal to ML (since KM bisects the side KL).

we will use the given information that KM bisects ∠JKN and KN bisects ∠MKL.
Since KM bisects ∠JKN, it means that it divides ∠JKN into two equal angles.

So, ∠JKM = ∠KJN. (Definition of angle bisector)
Similarly, since KN bisects ∠MKL, it divides ∠MKL into two equal angles.

So, ∠KJN = ∠NKL. (Definition of angle bisector)
Now, we can use the transitive property of equality: if ∠JKM = ∠KJN and ∠KJN = ∠NKL, then ∠JKM = ∠NKL.

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Answer:

76.93 cm²

Step-by-step explanation:

Area of semi-circle = (1/2) · π · r²

r = 7 cm

π = 3.14

Let's solve

(1/2) · 3.14 · 7² = 76.93 cm²

So, the area of the semi-circle is 76.93 cm²

The area of a rhombus with the given diagonals measure is 48 square units.

Given that, the length of two diagonals are 8 units and 12 units.

We know that, Area of rhombus = 1/2 × (product of the lengths of the diagonals)

Here, Area = 1/2 × 8 × 12

= 48 square units

Therefore, the area of a rhombus with the given diagonals measure is 48 square units.

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The given rational function is expressed as the difference between two simpler rational expressions: 1 / (x - 4) - 1 / (x - 3). This is the expression of the rational function as a difference between two simpler rational expressions.

To express the given rational function as a sum or difference of two simpler rational expressions, follow these steps:
Given rational function: 1 / (x - 4)(x - 3)
Step 1: Let the two simpler rational expressions be A / (x - 4) and B / (x - 3).
Step 2: Express the original function as a sum of these two expressions:
1 / (x - 4)(x - 3) = A / (x - 4) + B / (x - 3)
Step 3: Clear the denominators by multiplying both sides by (x - 4)(x - 3):
1 = A(x - 3) + B(x - 4)
Step 4: Solve for A and B by substituting convenient values for x. For example, set x = 4:
1 = A(4 - 3) + B(4 - 4) => A = 1
Now, set x = 3:
1 = A(3 - 3) + B(3 - 4) => B = -1
Step 5: Plug the values of A and B back into the simpler expressions:
1 / (x - 4)(x - 3) = 1 / (x - 4) - 1 / (x - 3)
So, the given rational function is expressed as the difference between two simpler rational expressions: 1 / (x - 4) - 1 / (x - 3).

To express the rational function 1/(x-4)(x-3) as a sum or difference of two simpler rational expressions, we can use partial fraction decomposition. First, we need to factor the denominator as (x-4)(x-3). Then we can write:
1/(x-4)(x-3) = A/(x-4) + B/(x-3)
where A and B are constants to be determined. To solve for A and B, we can multiply both sides of the equation by (x-4)(x-3) and simplify:
1 = A(x-3) + B(x-4)
Expanding and equating coefficients of x, we get:
0x + 1 = Ax + Bx - 3A - 4B
Simplifying and grouping like terms, we get a system of two equations in two variables:
A + B = 0  (coefficients of x^1)
-3A - 4B = 1 (coefficients of x^0)
Solving this system, we get:
A = 1/(4-3) = 1
B = -1/(4-3) = -1
Therefore, we can write:
1/(x-4)(x-3) = 1/(x-4) - 1/(x-3)
This is the expression of the rational function as a difference between two simpler rational expressions.

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The particular solution of the given differential equation is: y = 2x - 3/2. The correct option is (d).

To solve this, we can use the method of undetermined coefficients, since the right-hand side is a polynomial.

1. First, guess a form for the particular solution: yp(x) = Ax + B, where A and B are constants to be determined.

2. Compute the first and second derivatives:
  yp'(x) = A
  yp''(x) = 0

3. Substitute these derivatives and the guess for yp(x) into the given differential equation:
  0 + 3A + 2(Ax + B) = 4x + 3

4. Equate coefficients of x and the constant terms on both sides:
  2A = 4 (coefficient of x)
  3A + 2B = 3 (constant term)

5. Solve this system of equations:
  A = 2
  B = -3/2

6. Plug A and B back into the guess for the particular solution:
  yp(x) = 2x - 3/2

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The proportion of stay-at-home residents in arkansas is approximately 0.0166. Please note that the numbers used in this example are hypothetical

To estimate the proportion of stay-at-home residents in Arkansas, you can follow these steps:

1. Find relevant data: Look for a reliable source that provides the necessary information about stay-at-home residents in Arkansas. This could be government reports, research studies, or online databases.

2. Identify the total population: Determine the total number of residents in Arkansas. According to the U.S. Census Bureau, the population of Arkansas in 2020 was around 3,011,524.

3. Identify the number of stay-at-home residents: From the data source, find the number of stay-at-home residents in Arkansas.

4. Calculate the proportion: To find the proportion, divide the number of stay-at-home residents by the total population of Arkansas. For example, if there are 50,000 stay-at-home residents in Arkansas, the proportion would be:

Proportion = (Number of stay-at-home residents) / (Total population)
Proportion = 50,000 / 3,011,524

5. Round to four decimal places: If required, round the resulting proportion to four decimal places. In our example:

Proportion ≈ 0.0166

Please note that the numbers used in this example are hypothetical, and you will need to find the actual number of stay-at-home residents in Arkansas from a reliable source to get the correct proportion.

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The events a and b in a sample space with  positive probability shows that  p(b|a) > p(b)when p(a|b) > p(a).

We want to prove that P(B|A) > P(B) if and only if P(A|B) > P(A) whic are positive probability.

First, let's recall the definition of conditional probability: P(B|A) = P(A ∩ B) / P(A) and P(A|B) = P(A ∩ B) / P(B).

Now, let's prove both directions of the statement:

(1) If P(B|A) > P(B), then P(A|B) > P(A):

Given that P(B|A) > P(B), we have:

P(A ∩ B) / P(A) > P(B)

Now, multiply both sides by P(A):

P(A ∩ B) > P(A) * P(B)

Now, divide both sides by P(B):

P(A ∩ B) / P(B) > P(A)

Thus, P(A|B) > P(A).

(2) If P(A|B) > P(A), then P(B|A) > P(B):

Given that P(A|B) > P(A), we have:

P(A ∩ B) / P(B) > P(A)

Now, multiply both sides by P(B):

P(A ∩ B) > P(A) * P(B)

Now, divide both sides by P(A):

P(A ∩ B) / P(A) > P(B)

Thus, P(B|A) > P(B).

Therefore, we have proven that P(B|A) > P(B) if and only if P(A|B) > P(A).

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To find the area under the standard normal curve to the left of z=−0.84, we need to use a standard normal distribution table or calculator.

Using a calculator, we can input the command "normalcdf(-999, -0.84)" (where -999 represents negative infinity) to find the area under the curve to the left of z=−0.84. This gives us a result of approximately 0.2005. Rounding this answer to four decimal places as requested, we get the final answer of 0.2005.

Therefore, the area under the standard normal curve to the left of z=−0.84 is 0.2005, you'll find the corresponding area to be approximately 0.2005. So, the area under the curve to the left of z = -0.84 is approximately 0.2005, rounded to four decimal places.

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The unique solution to the given system is x(t) = [tex]e^t[/tex][1; -1] + [tex]e^5^t[/tex][1; 1].

To solve the given 2x2 system of differential equations x'(t) = Ax(t) with the initial condition x(0) = [2; 0], we find the eigenvalues and eigenvectors, and then express the solution as x(t) = [tex]e^(^d^1^t^)[/tex]v1 + [tex]e^(^d^2^t^)[/tex]v2.

1. Find the matrix A: A = [3, 2; 2, 3]
2. Find eigenvalues (d1, d2) and eigenvectors (v1, v2) of A.
3. Calculate the matrix exponential using the eigenvalues and eigenvectors.
4. Apply the initial condition x(0) = [2; 0] to find the coefficients.

Following these steps, we find d1 = 1, d2 = 5, v1 = [1; -1], and v2 = [1; 1].

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The differential equation for the amount of pollutant y(t) in the lake is dy/dt = 550/yr - (y(t)/12200)(67.1 km^3/yr), where y(t) is measured in tons and t is measured in years. The solution to the differential equation is y(t) = (550/67.1)(1 - exp((-67.1/12200)t)) + 200000. As t goes to infinity, y(t) approaches 20818.5 tons.

The change in pollutant in the lake over a small time interval is given by the difference between the amount that enters the lake and the amount that leaves, plus the amount that is added directly:

dy/dt = (rate in) - (rate out) + (rate added)

The rate in and rate out are both equal to 67.1 km^3/yr, so we can substitute these values:

dy/dt = 550/yr - (y(t)/12200)(67.1 km^3/yr)

To solve the differential equation, we can use separation of variables:

dy/dt + (67.1/12200)y = 550/12200

Multiplying both sides by the integrating factor exp((67.1/12200)t), we get:

exp((67.1/12200)t)dy/dt + (67.1/12200)y exp((67.1/12200)t) = (550/12200)exp((67.1/12200)t)

This can be written as:

d/dt (exp((67.1/12200)t)y) = (550/12200)exp((67.1/12200)t)

Integrating both sides with respect to t,

exp((67.1/12200)t)y = (550/67.1)exp((67.1/12200)t) + C

where C is the constant of integration. We can find the value of C using the initial condition y(0) = 200000:

exp(0) * 200000 = (550/67.1)exp(0) + C

C = 200000 - (550/67.1)

Substituting this value back into the equation, we get:

exp((67.1/12200)t)y = (550/67.1)exp((67.1/12200)t) + 200000 - (550/67.1)

y(t) = (550/67.1)(1 - exp((-67.1/12200)t)) + 200000

As t goes to infinity, the exponential term exp((-67.1/12200)t) goes to zero, so y(t) approaches the steady state solution given by:

y(t) → (550/67.1) + 200000 ≈ 20818.5

In practical terms, this means that over time the amount of pollutants in the lake will approach a constant value of approximately 20818.5 tons. The rate at which the pollutants enter and leave the lake is balanced by the rate at which they are added directly, resulting in a steady state concentration of pollutants in the lake.

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According to the property of parallelogram, Angle ABC is right angle.

Given:In □ABCD is quadrilateral

AB=DC andAD=BC

To prove: Angle ABC is right angle

Proof: in △ABC and △ADC

AD=BC [Given]

AB=DC [Given]

AC=AC [Common side]

thus, By SSS property △ADC≅△ACB

∴∠DAC=∠DCA

∴AB∣∣DC

In△ABD and △DCB

DB=DB [Common side]

AD=BC [Given]

AB=DC [Given]

Thus, △ABD≅△DCB

AD∣∣BC

Since AB∣∣DC and AD∣∣BC, △ABCD is parallelogram

Therefore, Angle ABC is right angle.

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An eigenvalue for matrix a with eigenvector v, then u(t) eλtv is a solution to the differential du equation = a = au.  This is correct.

If λ is an eigenvalue for matrix A with eigenvector v, then Av = λv. Taking the derivative of u(t)v with respect to t, we get:

du/dt [tex]\times[/tex] v = u(t) [tex]\times[/tex] d/dt(v) = u(t) [tex]\times[/tex] Av = u(t) [tex]\times[/tex]λv

On the other hand, we have:

Au = λu

Multiplying both sides by v, we get:

Avu = λuv

Since v is nonzero (by definition of eigenvector), we can divide both sides by v to get:

Au = λu

So, du/dt [tex]\times[/tex]v = u(t) [tex]\times[/tex]λv = Au(t)v = Au(t)[tex]\times[/tex] (u(t)^(-1)v)

Since u(t)^(-1)v is just a scalar, say c, we have:

du/dt [tex]\times[/tex] v = λc[tex]\times[/tex] u(t)v

Therefore, u(t)v is a solution to the differential equation du/dt = Au, with eigenvalue λ.

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