show that a strictly diagonally dominant matrix is invertible. show furthermore that if all the diagonal entries are positive, then all the eigenvalues have positive real part.

Answer:

∠DCB = 32

Step-by-step explanation:

∠A + ∠B = 90

∠B = 90 - ∠A = 90 - 32 = 58

∠DCB + ∠B = 90

∠DCB = 90 - ∠B = 90 - 58 - 32

Answer:

Let's use the formula for the sum of the first n terms of an arithmetic sequence to find the value of the common difference d:

S = n/2 * (2a1 + (n-1)d), where S is the sum of the first n terms, a1 is the first term, and d is the common difference.

We know that S = 1830 and n = 30, so we can write:

1830 = 30/2 * (2a1 + 29d)

1830 = 15(2a1 + 29d)

122 = 2a1 + 29d

Next, we know that the 8th term is 31, so we can write:

a8 = a1 + 7d = 31

Now we have two equations with two unknowns, so we can solve for a1 and d:

122 = 2a1 + 29d

31 = a1 + 7d

Multiplying the second equation by 2 and subtracting it from the first equation, we get:

60 = 15d

So d = 4.

Substituting d = 4 into the equation a1 + 7d = 31, we get:

a1 + 28 = 31

a1 = 3

Therefore, the first three terms of the arithmetic sequence are:

a1 = 3

a2 = a1 + d = 3 + 4 = 7

a3 = a2 + d = 7 + 4 = 11

So the first three terms are 3, 7, 11.

In this case, we are given a car dealership that offers a convertible that can be purchased in one of four colors: red, black, white, or silver. It is asked in the problem to give the probability for the next car purchased will be silver. since there are 300 silver colors, then the probability is 300/1500 or 1/5 in lowest terms. this is equal to 0.20.

The total number of cats at the vet is approximately 61.

A percentage is an equation proving the equality of two ratios. A ratio is a fractional comparison of two numbers or quantities. Proportions are used in mathematics to solve issues that involve comparing two numbers or discovering an unknown value. For instance, proportions can be used to determine equivalent fractions, compute percentages, and solve issues requiring rates and ratios. In a proportion, the numerator of one ratio is the same as the numerator of the other ratio, and vice versa for the denominator. Simplifying and cross multiplying can be used to address proportional problems.

Let the total number of dogs =x.

Let the total number of cats = y.

Given that, for every seven dogs at the vet there are 10 cats.

Thus, using proportions we have:

7 dogs / 10 cats = x / y

Using cross multiplication:

7 dogs x y = 10 cats (x)

y = (10/7) (x)

Now, x +y = 102

x = 102 - y

Substituting the value:

y = 10/7 (102 - y)

y = 145.71 - 1.4y

y + 1.4y = 145.71

2.4y = 145.71

y = 60.71

Hence, the total number of cats at the vet is approximately 61.

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The value of ML in the given quadrilateral which is a trapezoid is 58 units.

A polygon with only one set of parallel sides is called a trapezoid. The parallel bases of a trapezoid are another name for these parallel sides. Trapezoids have two additional sides that are not parallel and are referred to as their legs.

Trapezoids are defined differently by different people. A trapezoid can have one and only one pair of parallel sides, according to one school of mathematics, whereas another contends that a trapezoid can have several pairs of parallel sides. If we take into account the second definition, then a parallelogram is also a trapezoid under that definition.

We know that, A median on a trapezoid will be parallel to the bases, with a length equal to the sum of the bases divide by 2.

Thus,

45 = 3x + 11 + 10x - 12 / 2

45 = 13x - 1/ 2

90 = 13x - 1

91 = 13x

x = 7

Substitute the value of x in ML:

ML = 10x - 12

ML = 10(7) - 12

ML = 70 - 12

ML = 58

Hence, the value of ML in the given quadrilateral which is a trapezoid is 58 units.

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New points of graph A'B'C'D' are A'(-2, -2), B'(-2, 0), C'(-4, 0), D'(-4, -1)

In graph theory, the term "translation" refers to a type of operation that moves all the vertices and edges of a graph by a fixed distance in a given direction. Specifically, a translation of a graph involves shifting every vertex a certain distance horizontally and/or vertically, without changing the shape or connectivity of the graph.

Translation: 4 left and 2 down

Start with a point at its original location and then move it 4 units to the left and 2 units down. This can be done by subtracting 4 from the x-coordinate and subtracting 2 from the y-coordinate of the point or shape.

Given points in a graph ABCD are, A(2, 0), B(2, 2), C(0, 2), D(0, 1)

Subtract 4 from the x-coordinate and subtract 2 from the y-coordinate, resulting in a new points of graph A'B'C'D' are A'(-2, -2), B'(-2, 0), C'(-4, 0), D'(-4, -1)

The figure shown in below diagram.

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Therefore, the length of the diagonal of the rectangle is approximately 193.57 cm or 49.43 cm, depending on which value of x is used.

Perimeter is the total distance around the edge of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape. For example, the perimeter of a rectangle is found by adding the lengths of its four sides.

Here,

(i) Let the length of the rectangle be y cm.

Then, the perimeter of the rectangle is given by:

2(x + y) = 112

x + y = 56

y = 56 - x

(ii) The area of the rectangle is given by:

Area = length x breadth

597 = yx

Substituting y = 56 - x, we get:

597 = x(56 - x)

597 = 56x - x²

x² - 56x + 597 = 0

(iii) Using the quadratic formula,

x = (-(-56) ± √((-56)² - 4(1)(597))) / (2(1))

x = (56 ± √(3136 - 2388)) / 2

x = (56 ± √(748)) / 2

x = (56 ± 2√(187)) / 2

x = 28 ± √(187)

Therefore, the two solutions are x = 28 + √(187) and x = 28 - √(187).

(iv) The length of the rectangle is y = 56 - x.

Using Pythagoras theorem, the length of the diagonal of the rectangle is given by:

d² = y² + x²

d² = (56 - x)² + x²

d² = 3136 - 112x + 2x²

d = √(3136 - 112x + 2x²)

Substituting the value of x from part (iii) into the above equation, we get:

d = √(3136 - 112(28 ± √(187)) + 2(28 ± √(187))²)

d = √(3136 - 3136 ± 112√(187) + 56 ± 56√(187) + 2(187))

d = √(37400 ± 168√(187))

d ≈ 193.57 cm (rounded to 2 decimal places) or d ≈ 49.43 cm (rounded to 2 decimal places)

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Form refers to three-dimensional objects with depth, while shape pertains to the two-dimensional outline or boundary of an object.

We have,

In the context of geometry and visual representation, the terms "form" and "shape" have distinct meanings and characteristics.

Form generally refers to a three-dimensional object that has depth, such as a solid object or a structure with volume.

It encompasses objects that have length, width, and height, and it extends beyond a two-dimensional representation.

Form can have irregular or complex shapes and is not limited to a specific number of sides.

Shape, on the other hand, refers to the two-dimensional outline or boundary of an object.

It is limited to the external appearance or silhouette of an object without considering its depth or volume.

Shapes are typically described by their attributes, such as the number of sides (e.g., triangle, square) or specific geometric properties (e.g., circle, rectangle).

Thus,

Form refers to three-dimensional objects with depth, while shape pertains to the two-dimensional outline or boundary of an object.

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To solve the question asked, you can say:  So, the other angle of the figure is 49 degree.

In Euclidean geometry, an angle is a shape consisting of two rays, known as sides of the angle, that meet at a central point called the vertex of the angle. Two rays can be combined to form an angle in the plane in which they are placed. Angles also occur when two planes collide. These are called dihedral angles. An angle in planar geometry is a possible configuration of two rays or lines that share a common endpoint. The English word "angle" comes from the Latin word "angulus" which means "horn". A vertex is a point where two rays meet, also called a corner edge.  

here the given angles are as -

107 + (180-156) + x = 180

as total angle sum of a triangle is 180

so,

x = 180 - 131

x = 49

So, the other angle of the figure is 49 degree.

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At the end of the year, the expression that shows the total number of students is [tex]22+s[/tex]

An expression is a mathematical phrase that can contain numbers, variables, and operators. It doesn't contain an equal sign (=) or a value.

Mrs. Jeffers started the school year with 22 students. During the school year, another s student joined her class.

The expression that shows the number of students at the end of the year is.[tex]22+s[/tex]

The symbol "+" is an operator that stands for addition, and "s" is a variable that represents the number of students who joined the class.

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According to the solution, the Given number is not prime (7/8 or 0.875). This doesn't qualify as a prime number.

Prime numbers are those that have just two elements, one and themselves. For instance, the first five prime numbers are 2, 3, 5, 7, and 11. Comparatively, composite numbers are defined as having more than two elements.

Try to divide the number by all the smaller numbers to see if your child can determine whether it is prime. It is a prime number if it can only be divided by itself and by one.

= 4/7+1/8+1/3

=  7/8 or 0.875

This is not a prime number.

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The missing cοοrdinate r is 11

Slοpe is calculated by finding the ratiο οf the "vertical change" tο the "hοrizοntal change" between (any) twο distinct pοints οn a line. Sοmetimes the ratiο is expressed as a quοtient ("rise οver run"), giving the same number fοr every twο distinct pοints οn the same line.

First use pοint slοpe fοrm tο find yοur equatiοn

y - y1 = m ( x - x1) using (-6,3) and m= 1/2

y - 3 = 1/2 ( x - -6)

y - 3 = 1/2x + 3

Add 3 tο each side tο isοlate y

y -3 +3 = 1/2x +3 +3

y = 1/2x + 6

Nοw we knοw οur y-intercept is 6

Sο we can graph (-6, 3) and (0, 6) and draw a line

Nοw draw a vertical line at x = 10 because we want tο find the y value at x=10

y = 11

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See attached graph

Let Y be a binomial random variable with n trials and probability of success given by p. We are to show that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p.

Binomial random variables are a type of discrete random variables which have the following properties:Each trial is independent, with two possible outcomes called success and failureThe probability of success p is the same for each trial. The binomial random variable X is the number of successes that occur in n trials.

The probability distribution of X is given by:P(X = x) = ( n choose x ) p^x(1 - p)^(n - x)for x = 0, 1, 2, ... , nThe moment-generating function of X is given by:M(t) = E(e^(tX)) = sum_(x=0)^n [ (n choose x) p^x(1-p)^(n-x) e^(tx)]We are to show that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p.

Since Y is a binomial random variable, it has the following moment-generating function:M_Y(t) = E(e^(tY)) = sum_(y=0)^n [ (n choose y) p^y(1-p)^(n-y) e^(ty)]Note that U = n - Y, so that n - u = y or u = n - y. Then the moment-generating function of U is given by:M_U(t) = E(e^(tU)) = E(e^(t(n - Y))) = E(e^(nt) e^(-tY)) = e^(nt) E(e^(-tY)) = e^(nt) M_Y(-t).

Thus the moment-generating function of U is the product of a term that is independent of y and the moment-generating function of a binomial random variable with n trials and probability of success 1-p. Therefore, U is a binomial random variable with n trials and probability of success 1-p.

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Answer: $318/mo

Step-by-step explanation:

First, we get the remainder, which is the difference between $9, 632 and $2,000. That gives us $7, 632.

Then, since we know she will pay in 24 months, we assume she pays the same amount each month and divide $7, 632 by 24 = $318/mo

The time in hours which it will take them for working together for terri is about 1.71 hours.

Inverse Proportion: When two quantities are related in such a way that the product of one quantity with the reciprocal of the other quantity remains constant, it is said to be in inverse proportion.

Let's calculate their working rate:

Terri takes 3 hours to complete the painting of her living room, so she can paint her living room in [tex]\frac{1}{3} hours[/tex]

Angela takes 4 hours to complete the painting of the living room, so she can paint her living room in [tex]\frac{1}{4} hours[/tex]

If both work together, then the time taken to complete the work will be less than the time taken by each of them individually.

To find the time taken by both working together, we will add their rates.

Terri's work rate = [tex]\frac{1}{3} hours[/tex]

Angela's work rate = [tex]\frac{1}{4} hours[/tex]

Work rate when working together = Terri's rate + Angela's rate= [tex]\frac{1}{3} + \frac{1}{4} = \frac{7}{12}[/tex]

Thus, both will take [tex]\frac{12}{7} = 1.71 hours[/tex] approximately to complete the painting of the living room when they work together.

Therefore, the time in hours is about 1.71 hours.

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Answer:

2

Step-by-step explanation:

Seq. nth term, sum

The nth term of a sequence is n^2+ a

The 6th term of the sequence is 29

Find the sum of the first 4 terms

We are given that the nth term of the sequence is n^2 + a.

To find the value of 'a', we can use the fact that the 6th term of the sequence is 29.

Substituting n = 6 in the expression for the nth term, we get:

6^2 + a = 29

Simplifying this equation, we get:

a = 29 - 6^2

a = -7

So, the expression for the nth term of the sequence is:

n^2 - 7

Now, we need to find the sum of the first 4 terms of the sequence.

The first term of the sequence is given by substituting n = 1 in the expression for the nth term:

1^2 - 7 = -6

The second term of the sequence is given by substituting n = 2:

2^2 - 7 = -3

The third term of the sequence is given by substituting n = 3:

3^2 - 7 = 2

The fourth term of the sequence is given by substituting n = 4:

4^2 - 7 = 9

Therefore, the sum of the first 4 terms of the sequence is:

-6 + (-3) + 2 + 9 = 2

The exponential equation that fits the data points (0,35), (1,50), (2,100), (3,200), and (4,400) is y = 35 * (10/7)^x.

To find an exponential equation that fits the given data points, we can use the general form of an exponential equation:

y = a * b^x

where y is the dependent variable (in this case, the second coordinate of each data point), x is the independent variable (the first coordinate of each data point), a is the initial value of y when x is 0, and b is the growth factor.

Using the given data points, we can create a system of equations:

35 = a * b^0

50 = a * b^1

100 = a * b^2

200 = a * b^3

400 = a * b^4

The first equation tells us that a = 35, since any number raised to the power of 0 is 1. We can then divide the second equation by the first equation to get:

50/35 = b^1

Simplifying, we get:

10/7 = b

We can now substitute a = 35 and b = 10/7 into the remaining equations and solve for y:

y = 35 * (10/7)^x

This is the exponential equation that fits the given data points. We can use it to find the value of y for any value of x. This equation gives us a way to predict the value of y for any value of x.

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There were 0.00325 moles of oxygen gas (O2) involved in the reaction.

The magnesium oxide created will react with any oxygen in the air when the experiment is performed without a lid on the crucible.

Using the atomic mass of magnesium, we can calculate the number of moles of oxygen gas (O2) that reacted by first calculating the number of moles of magnesium that reacted:

Magnesium mass (Mg) = 0.12 g

Mg's atomic mass is 24.31 g/mol (from periodic table)

Mg's mass divided by its atomic mass yields the number of moles.

= 0.12 g / 24.31 g/mol

= 0.00494 mol

The chemical equation for the reaction of magnesium and oxygen to form magnesium oxide is 2Mg + O2 2MgO.

According to the equation, 2 moles of magnesium and 1 mole of oxygen combine to form 2 moles of magnesium oxide.

Thus, the following formula can be used to determine how many moles of O2 reacted:

Production of MgO moles as a function of mass

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The correct answer is: (4, 3) is a member. Its cardinality is 4.

Matching the sets on the left with a true statement about the Cartesian product of those sets on the right:{1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}{1, 2, 3, 4} × {3, 4, 5, 6} = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 3), (4, 4), (4, 5), (4, 6)}{4, 5, 6, 7} × {4, 5, 6, 7} = {(4, 4), (4, 5), (4, 6), (4, 7), (5, 4), (5, 5), (5, 6), (5, 7), (6, 4), (6, 5), (6, 6), (6, 7), (7, 4), (7, 5), (7, 6), (7, 7)}{a, e, i, o, u} × {b, g, t, d} = {(a, b), (a, g), (a, t), (a, d), (e, b), (e, g), (e, t), (e, d), (i, b), (i, g), (i, t), (i, d), (o, b), (o, g), (o, t), (o, d), (u, b), (u, g), (u, t), (u, d)}{1, 2, 3} × {1, 2, 4} = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (3, 4)}The following are true statements about the Cartesian product of these sets:its cardinality is 4. (4, 3) is a member.

Therefore, the correct answer is: (4, 3) is a member. Its cardinality is 4.

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The segment drawn from the apex to the circumference of the circle's base is 25 units long.

Let's denote the radius of the circular base of the cone by r, and the height of the cone by h. Then, the diameter of the base is given as 14 units, which means that the radius is r = 14/2 = 7 units.

We are given that the volume of the cone is 392π cubic units, which means that:

(1/3)πr²h = 392π

Simplifying this equation, we get:

r²h = 1176

Substituting r = 7, we get:

49h = 1176

Solving for h, we get:

h = 24

So the height of the cone is 24 units.

To find the length of a segment drawn from the apex to the edge of the circular base, we can use the Pythagorean theorem. Let's denote this length by L. Then, we have:

L² = r² + h²

Substituting r = 7 and h = 24, we get:

L² = 7² + 24²

L² = 625

Taking the square root of both sides, we get:

L = 25

Therefore, the length of the segment drawn from the apex to the edge of the circular base is 25 units.

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Answer: The given equation 2x + 5y = 10 can be rewritten in slope-intercept form (y = mx + b) by solving for y:

2x + 5y = 10

5y = -2x + 10

y = (-2/5)x + 2

where the slope is -2/5.

Since we want to find the equation of a line parallel to this one, the slope of the new line will also be -2/5. We can use the point-slope form of the equation of a line to find the equation of the new line, using the point (0,-3):

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point.

Substituting m = -2/5, x1 = 0, and y1 = -3, we get:

y - (-3) = (-2/5)(x - 0)

y + 3 = (-2/5)x

y = (-2/5)x - 3

Therefore, the equation of the line parallel to 2x + 5y = 10 which passes through (0,-3) is y = (-2/5)x - 3.

Step-by-step explanation:

Answer:

2x + 5y = - 15

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

2x + 5y = 10 ( subtract 2x from both sides )

5y = - 2x + 10 ( divide through by 5 )

y = - [tex]\frac{2}{5}[/tex] x + 2 ← in slope- intercept form

with slope m = - [tex]\frac{2}{5}[/tex]

• Parallel lines have equal slopes , then

y = - [tex]\frac{2}{5}[/tex] x + c

the line crosses the y- axis at (0, - 3 ) ⇒ c = - 3

y = - [tex]\frac{2}{5}[/tex] x - 3 ← equation of parallel line in slope- intercept form

multiply through by 5 to clear the fraction

5y = - 2x - 15 ( add 2x to both sides )

2x + 5y = - 15 ← in standard form

A blueprint shows an apartment with an area of 15 square inches. If the blueprint's scale is 1 inch : 8 feet. The actual area of the apartment will be 960 square feet.

Dilation:

Inflation is the process of increasing the size of an item without affecting its shape. Depending on the scale factor, the size of the object can increase or decrease. Dilation is a transformation used to change the size of an object. Dilation is used to make objects larger or smaller. This transformation produces an image of the same shape as the original.

The dilation should either stretch or contract the original shape. This transformation is referred to as the "scaling factor".

When zooming in produces a larger image, it is called zooming in.

If dilation produces a smaller image, this is called downscaling.

There is no effect of dilation on the angle.

A blueprint shows an apartment with an area of 15 square inches.

If the blueprint's scale is 1 inch : 8 feet.

Then the scale factor will be 8/1 feet per inch.

Then the actual area square footage of the apartment will be

Actual area = 15 x (scale factor)²

Then the actual area of the footage will be

⇒ Actual area = 15 x (8/1)²

⇒ Actual area = 15 x 64

⇒ Actual area = 960 square feet

Thus, the actual area of the apartment will be 960 square feet.

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Answer: x=1

Step-by-step explanation:

Vertical angles mean they are equal to each other. So first you would set the two equations equal to each other.

7x+26=4x+29

Then subtract 4x from both sides.

7x+26−4x=29

combine 7x and −4x to get 3x.

3x+26=29

Then subtract 26 from both sides.

3x=29−26

subtract 26 from 29 to get 3.

3x=3

Lastly divide both sides by 3.

3/3x=3/3

Divide 3 by 3 to get 1.

x=1

The amount that would have had to be invested 15 years ago and compounded at 4.2% compounded continuously to grow to the present value is $37,537.19.

The investment can be treated as a continuous stream with the interest paid at a rate of 4.2% compounded continuously.The present value of the investment can be calculated using the formula:P = C/e^(rt)Where,P = Present ValueC = Cash Flowsr = Interest Rate Per Periodt = Number of Periodse = Euler’s numberThe given values are as follows:C = $8,000 per yearr = 4.2% compounded continuously for 15 years.

C = $8,000e = 2.71828t = 15 yearsNow, we need to calculate the present value using the above formula.P = 8000/e^(0.042 x 15) = $82,273.24.The formula to calculate the amount that would have been invested 15 years ago is:A = P x e^(rt)Where,A = Future Value of the investmentP = Present Value of the investmentr = Rate of Interest Per Periodt = Number of Periodse = Euler’s numberThe present value of the investment is $82,273.24.

The rate of interest is 4.2% compounded continuously.t = 15 yearsNow, we need to calculate the amount that would have been invested 15 years ago.A = 82,273.24 x e^(0.042 x 15) = $37,537.19

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Answer:

x=4

Step-by-step explanation:

100

40

X

60

Y

+160

Ty

40

3

Answer:

x = 10

y = -9

Step-by-step explanation:

x = -3y - 17

2x + 3y = -7

Plug in x

2 (-3y - 17) + 3y = -7

-6y - 34 + 3y = -7

-3y = 27

y = -9

Plug in the value you got for y back into the equation to find the x value

x = -3(-9) - 17

x = 10

If the expression 1/2x was placed in the form ax^b where a and b are real numbers, then a + b equal to option (4) -1/2

The given expression is 1/(2x), which can be rewritten as:

1/(2x) = 1/2 × (1/x)

Here, we can see that the expression can be written in the form of ax^b, where a = 1/2 and b = -1.

To see why a = 1/2, notice that 1/2 is the coefficient of (1/x). And to see why b = -1, note that x^(-1) is the exponent on the variable x

So, we have:

1/(2x) = (1/2) × x^(-1)

And, a + b = (1/2) + (-1)

Add the numbers

= -1/2.

Therefore, the correct option is (4) -1/2.

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The parabolic trajectory of the falling prey can be described by the equation y = 228 – x2/57, where y is the height above the ground and x is the horizontal distance traveled in meters. In this case, the prey was dropped at a height of 228 m and flying at 19 m/s. To calculate the total distance traveled by the prey, we can use the equation for the parabola to solve for x.

We can rearrange the equation y = 228 – x2/57 to solve for x, which gives us[tex]x = √(57*(228 – y))[/tex]. When the prey hits the ground, the height (y) is 0. Plugging this into the equation for x, we can calculate that the total distance traveled by the prey is[tex]x = √(57*(228 - 0)) = √(57*228) = 84.9 m.\\[/tex] Expressing this answer to the nearest tenth of a meter gives us the final answer of 84.9 m.

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The probability that the first ball is yellow if the second ball is black is 1/14. The correct option is D.

The given question is a classic example of dependent events in probability. As the balls are drawn without replacement, the second event's outcome will depend on the outcome of the first event.

Probability = Number of favorable events/ Total number of events

The probability of the first ball being yellow is [tex](5/15)[/tex], while the probability of the second ball being black is [tex](3/14)[/tex].

Mathematically represented as P(Yellow ball on the first draw) = P(Yellow ball) = [tex]5/15[/tex]

P(Blackball on second draw given Yellow ball on the first draw) = P(Blackball | Yellow ball) = [tex]3/14[/tex]

As both the events are dependent, we need to find the joint probability of both the events, which can be calculated as P(Yellow ball on the first draw and Blackball on the second draw) = P(Yellow ball) × P(Blackball | Yellow ball)

P (Yellow ball on the first draw and blackball on second draw) = [tex](5/15) × (3/14) = 3/42 = 1/14.[/tex]

Therefore, the correct option is D.

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U is a binomial random variable with n trials and probability of success given by 1 - p.

As Y is a binomial random variable with n trials and probability of success given by p. Using the moment-generating functions method, it can be shown that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p. The binomial distribution is described by two parameters: n, which is the number of trials, and p, which is the probability of success in any given trial. If a binomial random variable is denoted by Y, then:[tex]P(Y = k) = \binom{n}{k}p^{k}(1 - p)^{n-k}[/tex]

The method of generating moments can be used to show that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p. The moment-generating function of a binomial random variable is given by: [tex]M_{y}(t) = [1 - p + pe^{t}]^{n}[/tex]

The moment-generating function for U is: [tex]M_{u}(t) = E(e^{tu}) = E(e^{t(n-y)})[/tex]

Using the definition of moment-generating functions, we can write: [tex]M_{u}(t) = E(e^{t(n-y)})$$$$= \sum_{y=0}^{n} e^{t(n-y)} \binom{n}{y} p^{y} (1-p)^{n-y}[/tex]

Taking the summation of the above expression: [tex]= \sum_{y=0}^{n} e^{tn} e^{-ty} \binom{n}{y} p^{y} (1-p)^{n-y}$$$$= e^{tn} \sum_{y=0}^{n} \binom{n}{y} (pe^{-t})^{y} [(1-p)^{n-y}]^{1}$$$$= e^{tn} (pe^{-t} + 1 - p)^{n}[/tex]

Comparing this expression with the moment-generating function for a binomial random variable, we can say that U is a binomial random variable with n trials and probability of success given by 1 - p.

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