Let V be a vector space over F, and fix y∈V. Show that g:V→F defined by g(x)=⟨x,y⟩ for all x∈V is linear.

Answer: ??

Step-by-step explanation:

The probability that the height of a randomly selected tree is as tall as mine or shorter is 0.1151. The probability that the full height of a randomly selected tree is at least as tall as hers is 0.0362.

To find the probability that the height of a randomly selected tree is as tall as yours or shorter, we need to find the z-score for your tree's height and use the standard normal table to find the corresponding probability. The z-score is calculated as follows:

z = (x - μ) / σ

where x is the observed value, μ is the mean, and σ is the standard deviation.

For your tree, the z-score is:

z = (44.7 - 56.6) / 9.9 = -1.198

Using the standard normal table, the probability that the height of a randomly selected tree is as tall as yours or shorter is 0.1151.

For your neighbor's tree, the z-score is:

z = (74.4 - 56.6) / 9.9 = 1.798

Using the standard normal table, the probability that the height of a randomly selected tree is at least as tall as hers is 1 - 0.9638 = 0.0362.

So, the probability that the height of a randomly selected tree is as tall as yours or shorter is 0.1151, and the probability that the height of a randomly selected tree is at least as tall as your neighbor's is 0.0362.

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a. The equation "(a) x2+3​=x+4" gives the solution as x = 1.

b. The equation "(b) 2x-1​-x-4=2" gives the solution as x = -1.

To solve these radical equations, we will need to isolate the radical on one side of the equation and then square both sides to get rid of the radical. After that, we can solve for x using standard algebraic methods.

(a) x2+3​=x+4

First, we will isolate the radical on one side of the equation:
x2+3​=x+4
x2=x+4-3
x2=x+1

Now, we will square both sides of the equation to get rid of the radical:
(x2)2=(x+1)2
x4=x2+2x+1

Next, we will rearrange the equation and set it equal to zero:
x4-x2-2x-1=0

Now, we can use the quadratic formula to solve for x:
x=(-(-2)±√((-2)2-4(1)(-1)))/(2(1))
x=(2±√(4+4))/(2)
x=(2±√8)/(2)
x=(2±2√2)/(2)
x=1±√2

So, the solutions for x are 1+√2 and 1-√2.

(b) 2x−1​−x−4​=2

First, we will isolate the radical on one side of the equation:
2x−1​=x−4+2
2x−1​=x−2

Now, we will square both sides of the equation to get rid of the radical:
(2x−1)2=(x−2)2
4x2-4x+1=x2-4x+4

Next, we will rearrange the equation and set it equal to zero:
3x2=3

Now, we can solve for x by dividing both sides of the equation by 3 and taking the square root:
x2=1
x=±√1
x=±1

So, the solutions for x are 1 and -1.

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It will take 13 years for the account value to reach $10,800.

Compound interest means addition of interest to the principal sum of a loan or deposit which means interest on principal plus interest.

We will use the formula for compound interest [tex]A = P(1 + r/n)^{nt}[/tex]

Given that

A = Final account value ($10,800)

P = Principal amount ($4,500)

r = Annual interest rate (7% or 0.07)

n = 1

t = ?

$10,800 = $4,500(1 + 0.07/1)^(1*t)

Dividing by $4,500:

2.4 = (1.07)^t

Taking the logarithm of sides:

log(2.4) = log((1.07)^t)

Using logarithmic identity log(a^b) = b*log(a):

log(2.4) = t*log(1.07)

Dividing by log(1.07):

t = log(2.4) / log(1.07)

t = 12.9394949072

t = 13.

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If the stone dropped by Maria falls 2 meter in 1st second, then the stone would have fallen 30 meter after 10 seconds.

The distance that the stone falls in the first second is = 2 meters;

The distance that the stone falls in the 50th second is = 149 meters;

So, the rate of fall of the stone can be calculated as:

⇒ (149 - 2)/(50 - 1);

⇒ 147/49;

⇒ 3.

So , the stone is falling at the rate of 3 meter per second,

The distance fall by the stone after 10 second can be calculated as :

⇒ 10×3

⇒ 30 meter.

Therefore, After 10 seconds the stone would have fallen 30 meter.

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The converse that can be used to prove that the given lines are parallel given each angle pair is:

1. Corresponding angles theorem

2. Same-side interior angles can be used

The converse of the corresponding angles theorem states that if a transversal intersects two lines, and the corresponding angles formed by the transversal and the lines are congruent, then the two lines are parallel.

The converse of the same-side interior angles theorem can be stated as if a transversal intersects two lines, and the sum of the interior angles on one side of the transversal is 180 degrees, then the two lines are parallel.

Therefore we can conclude that:

A. The converse of the corresponding angles theorem

B. The converse of the same-side interior angles can be used

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Therefore, the best statement supported by the information in the box plots is statement B, "the data for Mr. Flores’s class are more symmetrical than the data for Mrs. Morales's class."

A box plot, also known as a box-and-whisker plot, is a graphical representation of the distribution of numerical data. It displays the five-number summary of a dataset, which includes the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value.

Here,

From the box plots, we can make the following observations:

A) The range for Mrs. Morales's class appears to be 100-50 = 50.

The range for Mr. Florez's class appears to be 95-55 = 40.

Therefore, statement A is not supported by the information in the box plots.

B) The box plot for Mr. Flores's class is more symmetrical than the box plot for Mrs. Morales's class, which is skewed to the left.

Therefore, statement B is supported by the information in the box plots.

C) The median score for Mrs. Morales's class appears to be between 70 and 75.

The median score for Mr. Florez's class appears to be between 75 and 80.

Therefore, statement C is not supported by the information in the box plots.

D) The interquartile range for Mrs. Morales's class appears to be between 60 and 80, which is a range of 20.

The interquartile range for Mr. Florez's class appears to be between 65 and 85, which is also a range of 20.

Therefore, statement D is supported by the information in the box plots.

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Therefore, the value of angle A is approximately 76.16°, and the value of angle C is approximately 8.24°.

A triangle is a closed two-dimensional shape with three straight sides and three angles. It is the simplest polygon that can be formed in Euclidean geometry, and it is widely studied in mathematics and geometry. Triangles can be classified based on their sides and angles. By side length, a triangle can be classified as equilateral (all sides are equal), isosceles (two sides are equal), or scalene (no sides are equal). By angle measure, a triangle can be classified as acute (all angles are less than 90 degrees), obtuse (one angle is greater than 90 degrees), or right (one angle is exactly 90 degrees). Triangles have many interesting properties and are used in a variety of applications, including in architecture, engineering, and physics.

Here,

Sure, I can help you with that! We can use the Law of Cosines to find the values of angles A and C in triangle ABC. The Law of Cosines states that:

c² = a² + b² - 2ab cos(C)

where c is the side opposite angle C.

First, let's find the value of side c:

c² = a² + b² - 2ab cos(C)

c² = (782)² + (1435)² - 2(782)(1435) cos(115.6°)

c² = 613524 + 2055225 - 2238015.52

c²  = 594733.48

c = √(594733.48)

c ≈ 771.1 cm

Now, we can use the Law of Cosines again to find angle A:

cos(A) = (b² + c² - a²) / 2bc

cos(A) = (1435² + 771.1² - 782²) / (2 * 1435 * 771.1)

cos(A) = 0.2354

A = cos⁻¹*²³⁵⁴⁾

A ≈ 76.16°

Finally, we can find angle C by using the fact that the angles in a triangle add up to 180°:

C = 180° - A - B

C = 180° - 76.16° - 115.6°

C ≈ 8.24°

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The matrix \( A \) is not diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \), because there does not exist a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

The matrix \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1\end{array}\right] \) is diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \) if and only if there exists a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

To determine if \( A \) is diagonalizable, we need to find the eigenvalues and eigenvectors of \( A \).

The characteristic polynomial of \( A \) is given by:

\(\det(A-\lambda I)=\left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 0 & 1-\lambda & 1 \\ 0 & -1 & 1-\lambda\end{array}\right|=(1-\lambda)^{3} \)

The eigenvalues of \( A \) are the roots of the characteristic polynomial, which are \( \lambda=1 \) with multiplicity 3.

To find the eigenvectors of \( A \), we need to solve the equation \( (A-\lambda I)x=0 \) for each eigenvalue.

For \( \lambda=1 \), we have:

\( (A-1I)x=0 \)

\(\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{array}\right]x=0 \)

The solution to this equation is the eigenspace of \( \lambda=1 \), which is spanned by the eigenvector \( x=\left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right] \).

Since the eigenspace of \( \lambda=1 \) has dimension 1, there is only one linearly independent eigenvector for this eigenvalue. Therefore, the matrix \( A \) is not diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \), because there does not exist a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

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-13 is the  value of b in linear equation.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

                              Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

y = 6x + 4

 y = mx + c

is a straight line  of gradient of c.

  m = 6

y = 6x + c

the line passes through points(x,y) where x = 3 and y = 5

      5 = 6 * 3 + c

        c = 5 - 18

         c = -13

         y = 6x - 13

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If the extra large box have a dilation scale factor of 4, then the extra large box contain 76.8ounces.

If the extra large box is a dilation of the single serving box with a scale factor of 4, then

The ratio of the volumes of the extra large box to the single serving box will be 4³,

We know that, Volume is a three-dimensional measurement and is affected by the cube of the scale factor.

So, the volume of the extra large box will be;

⇒ 4³ × 1.2 ounces = 4×4×4×1.2 ounces = 76.8 ounces,

Therefore, the extra large box will contain 76.8 ounces of cereal.

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The volume of the larger cylinder as 70 m³.

What is a cylinder, simply defined?

With two parallel bases joined by a curving surface, a cylinder is a three-dimensional solid. The bases are typically shaped like circles. The cylinder's height ("h") and radius ("r") are used to represent the perpendicular distance between the bases.

The scale factor of the cylinders = 5:2

The scaling factor indicates how much a figure has increased or decreased from its initial value.

The volume of the smaller cylinder = 28 m³

We are asked to find the volume of the larger cylinder.

let the volume of the larger cylinder be x.

x / 28 = 5/2

x  =(5/2) * 28 = 70

Therefore using the scale factor, we found the volume of the larger cylinder as 70 m³.

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We can use LCM to find a common denominator for these fractions. The LCM of 70, 2, and 5 is 70 x 2 x 5 = 700. The result of the expression is approximately 29.3885.

To solve this problem using LCM, we first need to convert all the mixed numbers to improper fractions:

Three and three seventieths = (3 x 70 + 3) / 70 = 213 / 70

Six and a half = (6 x 2 + 1) / 2 = 13 / 2

Nine and three fifths = (9 x 5 + 3) / 5 = 48 / 5

Now we can use LCM to find a common denominator for these fractions. The LCM of 70, 2, and 5 is 70 x 2 x 5 = 700.

We can then convert each fraction to an equivalent fraction with denominator 700:

213/70 = 3.042857... ≈ 3.043

13/2 = 455/70 = 6.5

48/5 = 192/20 = 96/10 = 480/50

Now we can substitute these equivalent fractions into the original expression and simplify:

3.043 × 6.5 + 480/50 = 19.7885... + 9.6 = 29.3885.

LCM stands for the "Least Common Multiple" and is a mathematical concept used to find the smallest multiple that two or more numbers have in common. In other words, it is the smallest positive integer that is divisible by all the given numbers.

To find the LCM of two or more numbers, we can start by finding their prime factorization. Then, we can take the highest power of each prime factor that appears in any of the factorizations and multiply them together to get the LCM. Taking the highest power of each prime factor (2^3 x 3^1), we get 24, which is the smallest multiple that both 6 and 8 have in common.

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Quantitative, discrete, ratio

(i) Quantitative, discrete, interval
(ii) Qualitative, nominal
(iii) Qualitative, ordinal
(iv) Quantitative, discrete, ratio

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\[ \left[\begin{array}{rr}-35 & 7 \\ 5 & -1\end{array}\right] \]

In Exercises 1-12, use the Gauss-Jordan method to compute the inverse, if it exists, of the matrix.

1. \( \left[\begin{array}{ll}7 & 3 \\ 5 & 2\end{array}\right] \)

To find the inverse of this matrix, we need to use the Gauss-Jordan method. We begin by writing the augmented matrix:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
5 & 2 & 0 & 1
\end{array}\right]\]

Next, we subtract 5 times the first row from the second row to obtain:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
0 & -11 & -5 & 1
\end{array}\right]\]

Now, we divide the second row by -11 to obtain:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
0 & 1 & 5 & -1
\end{array}\right]\]

Finally, we subtract 7 times the second row from the first row to obtain:

\[ \left[\begin{array}{cc|cc}
1 & -20 & -35 & 7 \\
0 & 1 & 5 & -1
\end{array}\right]\]

Therefore, the inverse of the given matrix is:

\[ \left[\begin{array}{rr}-35 & 7 \\ 5 & -1\end{array}\right] \]

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Answer: 14.6

Step-by-step explanation:

The given sequence is A(n) = 6.3 + (n-1)(5), where n is the term number. To find the 5th term of the sixth term of the sequence, we need to first find the value of the sixth term and then find its fifth term.

To find the sixth term of the sequence, we substitute n = 6 in the given formula:

A(6) = 6.3 + (6-1)(5)

A(6) = 6.3 + 25

A(6) = 31.3

Now, to find the fifth term of this sequence, we need to go back one term from the sixth term, which means we need to substitute n = 5 in the original formula:

A(5) = 6.3 + (5-1)(5)

A(5) = 6.3 + 20

A(5) = 26.3

Therefore, the fifth term of the sixth term of the sequence A(n) = 6.3 + (n-1)(5) is 26.3.

The coordinates of the fourth vertex of the rectangle are (2, 5).

To find the fourth vertex of the rectangle, we need to first determine which sides of the triangle are opposite and parallel to each other. We can see that the two points (-6,3) and (-6,5) lie on a vertical line, which means they are opposite and parallel to each other.

Similarly, the two points (-6,3) and (2,3) lie on a horizontal line, which means they are also opposite and parallel to each other. The fourth vertex of the rectangle must lie at the intersection of these two lines.

The point (-6,3) is the lower left corner of the rectangle, and the point (2,3) is the lower right corner. The distance between these two points is 2 - (-6) = 8. Since the opposite sides of a rectangle are congruent, the distance between the other two corners (-6,5) and the fourth vertex must also be 8.

Since the two points (-6,3) and (-6,5) are on a vertical line, the y-coordinate of the fourth vertex must also be 5. We can find the x-coordinate by moving 8 units to the right of (-6,3), since the rectangle is parallel to the x-axis. Thus, the fourth vertex must be located at the point:

(x,y) = (-6+8, 5)

= (2, 5)

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Answer:

245

Step-by-step explanation:

The coordinates of the other endpoint, Q, are (3,-8).

The midpoint formula is used to find the coordinates of the midpoint of a line segment. The formula is as follows:
M = ((x1+x2)/2, (y1+y2)/2)

In this case, the midpoint M=(1,-2) and one endpoint P=(-1,4) are given. We can plug these values into the formula to find the coordinates of the other endpoint, Q:
(1,-2) = ((-1+x2)/2, (4+y2)/2)

Solving for x2 and y2, we get:
x2 = 2(1) + 1 = 3
y2 = 2(-2) - 4 = -8

Therefore, the coordinates of the other endpoint, Q, are (3,-8).

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Answer:

10:50 am, if you have to round the hour it would be 11 am on monday

Step-by-step explanation:

What I like to do for these type of problems is to add the 24hrs first so it would still be 9 am just on monday then we still have 1 hour and 50 minutes to add so

so it would be 10:50 am, if you have to round the hour it would be 11 am on monday

Answer:

10:50 am Monday

Step-by-step explanation:

Subtract 24 hours from 25 to skip a day.

25hours -24 hours=1 hour

It is monday 9am

Now we add 1 hour.

10 am monday

Add 50 minutes:

10:50 am Monday

Answer:

[tex]x \leqslant - 27[/tex]

Step-by-step explanation:

[tex]1. \: - \frac{x}{9} \geqslant 3 \\ 2. \: - x \geqslant 3 \times 9 \\ 3. \: - x \geqslant 27 \\ 4. \: x \leqslant - 27[/tex]

The basic subtraction of rational expression simplified is (7)/(12d).

To subtract these rational expressions, we need to find a common denominator. The least common denominator (LCD) of 4d and 6d is 12d. We can then rewrite the expressions with the LCD as the denominator:

(3)/(4d) = (3 * 3)/(4d * 3) = (9)/(12d)
(1)/(6d) = (1 * 2)/(6d * 2) = (2)/(12d)

Now we can subtract the numerators and keep the same denominator:

(9)/(12d) - (2)/(12d) = (9 - 2)/(12d) = (7)/(12d)

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 1:

(7)/(12d) = (7/1)/(12d/1) = (7)/(12d)

So the final answer is (7)/(12d).

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For two vectors to be orthogonal, their dot product should be zero.

(a) The dot product of u and v is k+21k=22k. For u and v to be orthogonal, 22k=0, which means k=0.

(b) The dot product of u and v is -2k^2+5k^2+k^2=4k^2. For u and v to be orthogonal, 4k^2=0, which means k=0.

Therefore, in both cases, the vectors are orthogonal when k=0.

Let AB and AC be vectors joining the points A(0,-1), B(1,-2), and C(4,1), respectively. Then,

AB = (1-0, -2-(-1)) = (1,-1)

AC = (4-0, 1-(-1)) = (4,2)

The cosine of the angle between two vectors u and v is given by the dot product of u and v divided by the product of their magnitudes, i.e., cos(theta) = (u.v)/(|u||v|).

(a) Cosine of angle A = cos(BAC) = (AB.AC)/(|AB||AC|) = (1(4) + (-1)(2))/sqrt(2^2+1^2 * 4^2+2^2) = 2/3

Cosine of angle B = cos(CAB) = (AC.AB)/(|AC||AB|) = (4(1) + 2(-1))/sqrt(4^2+2^2 * 2^2+1^2) = 1/3

Cosine of angle C = cos(ABC) = (AB.BC)/(|AB||BC|) = ((1)(3) + (-1)(-1))/sqrt(2^2+1^2 * 2^2+1^2) = 5/9

(b) Let AB and BC be vectors joining the points A(3,0,2), B(4,3,0), and C(8,1,-1), respectively. Then,

AB = (4-3, 3-0, 0-2) = (1, 3, -2)

BC = (8-4, 1-3, -1-0) = (4, -2, -1)

AC = (8-3, 1-0, -1-2) = (5, 1, -3)

The cosine of the angle between two vectors u and v is given by the dot product of u and v divided by the product of their magnitudes, i.e., cos(theta) = (u.v)/(|u||v|).

Cosine of angle A = cos(BAC) = (AB.AC)/(|AB||AC|) = (1(5) + 3(1) + (-2)(-3))/sqrt(1^2+3^2+(-2)^2 * 5^2+1^2+(-3)^2) = 0

Cosine of angle B = cos(CAB) = (AC.AB)/(|AC||AB|) = (5(1) + 1(3) + (-3)(-2))/sqrt(5^2+1^2+(-3)^2 * 1^2+3^2+(-2)^2) = -7/9

Cosine of angle C = cos(ABC) = (AB.BC)/(|AB||BC|) = ((1)(4) + (

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Answer:

1. 2430 feet

2. 1.8 m/s^2

Step-by-step explanation:

See the attached worksheet.

A time versus speed graph contains a small treasure of mathematical rewards. The area under the graph is equal to the distance travelled. The slope of the line segments represents acceleration.

For total distance: If we break the graph into three sections (2 triangles and a rectangle) we can calculate the areas for each. Each area is the distance travelled for that segment. As shown on the workseet, the total area is 2430 miles, the distance travelled by the train for this question.

The slope of the line in the first 10 seconds is 1.8 meters/sec^2, the acceleration of the train over that period.

Answer:

24 antelopes, 36 monkeys

Step-by-step explanation:

6 : 7 means that for every 6 antelopes, there are 7 monkeys, so

a/6 = m/7   (1)

where a is the number of antelopes, m is the number of monkeys (before the 8 were born).

After 8 monkeys are born, the new ratio becomes:

a/2 = (m+8)/3    (2)

You can solve a and m for these two equations:

Rewrite (1) as 7a = 6m (cross product)

Rewrite (2) as 3a = 2m+16  (cross product)

=> 2m = 3a-16 (isolate 2m)

=> 6m = 9a-48 (multiply by 3)

Plug in (2) into (1):

7a = 9a-48 =>

-2a = -48

a = 24 antelopes

m = (a/6)*7 = 28 monkeys BEFORE the 8 newborns, so now there are

24 antelopes and 28+8=36 monkeys

The expression can be simplified using the rule of exponents. The value of the expression 0.5a²b³ * (-2b)⁶ is 32a²b¹⁵.

The exponents can be added when multiplying two powers with the same base, according to the product rule of exponents. This rule may be expanded to encompass exponents that are negative or fractional, variables, constants, and more.

The expression can be simplified using the rule of exponents.

0.5a²b³ * (-2b)⁶ = 0.5a²b³ * 64b⁶

Using the power rule of the exponents we have:

= 32a²b⁹ * b⁶

Simplifying the above value we have:

= 32a²b¹⁵

Hence, the value of the expression 0.5a²b³ * (-2b)⁶ is 32a²b¹⁵.

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Expression 0.5a²b³ times (-2b)⁶ is -32a²b⁹

An algebraic expression is a combination of variables, constants, and mathematical operations. It represents a mathematical relationship between quantities and can be used to model real-world situations or solve problems.

Algebraic expressions can contain variables, which are symbols that represent unknown values, constants, which are fixed values, and mathematical operations such as addition, subtraction, multiplication, division, and exponents.

We can simplify this expression using the rules of exponents:

0.5a²b³ times (-2b)⁶ = 0.5a²b³ times (-64b⁶)

Multiplying the coefficients, we get:

-32a²b⁹

Therefore, the simplified expression is -32a²b⁹.

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arctan(4/7) + arctan(7/4) is 90 degrees

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and the functions that describe those relationships. It has applications in fields such as engineering, physics, astronomy, and navigation.

We can use the following trigonometric identity to simplify the expression:

arctan(x) + arctan(1/x) = pi/2

where x is a positive number.

In this case, we can set x = 4/7, which is a positive number, and apply the identity to get:

arctan(4/7) + arctan(7/4) = pi/2

Therefore, the sum of the two arctan terms simplifies to pi/2.

So, the simplified expression is:

arctan(4/7) + arctan(7/4) = 90 degrees (rounded to the nearest degree).

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Answer:

51.2

Step-by-step explanation:

By dividing :0

The probability that customer A is the last customer to leave the post office is e^(-2x).

The probability that customer A is the last customer to leave the post office can be found by using the formula for the exponential distribution, which is P(X > x) = e^(-λx). In this case, λ = 1 and x is the serving time for customer A.

First, we need to find the probability that customer A is served for a longer time than customer B. This can be found by subtracting the probability that customer A is served for a shorter time than customer B from 1. That is, P(X > x) = 1 - P(X < x).

P(X > x) = 1 - P(X < x) = 1 - (1 - e^(-λx)) = e^(-λx)

Next, we need to find the probability that customer A is served for a longer time than customer C. This can be found in the same way, by subtracting the probability that customer A is served for a shorter time than customer C from 1.

P(X > x) = 1 - P(X < x) = 1 - (1 - e^(-λx)) = e^(-λx)

Finally, we need to multiply these two probabilities together to find the probability that customer A is the last customer to leave the post office. P(A is the last customer to leave) = P(A is served longer than B) * P(A is served longer than C) = e^(-λx) * e^(-λx) = e^(-2λx)

Since λ = 1 and x is the serving time for customer A, the probability that customer A is the last customer to leave the post office is e^(-2x). Therefore, the probability that customer A is the last customer to leave the post office is e^(-2x).

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