Lisa records the amount of time she plans to spend for each reading session during a one month period on the line plot

5. If Lisa redistributes the total time
equally over all her reading sessions,
how long will each session be?

The concept of Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different theorems and axioms.

The concept of Euclidean geometry as required to be discussed is basically introduced for flat surfaces or plane surfaces. The postulates of the Euclidean geometry are as follows!

1 : A straight line may be drawn from any one point to any other point.

2 :A terminated line can be produced indefinitely.

3 : A circle can be drawn with any centre and any radius.

4 : All right angles are equal to one another (Congruent).

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The volume of the frustum of the pyramid is 4530 cubic units.

To find the volume of a frustum of a pyramid, we can use the formula

V = (1/3)h(A1 + A2 + sqrt(A1A2))

Where

V is the volume of the frustum.

h is the height of the frustum.

A1 is the area of the base of the frustum.

A2 is the area of the top of the frustum.

In this case, the frustum has a square base of side 27 and a square top of side 15. Therefore, we have

A1 = 27^2 = 729

A2 = 15^2 = 225

h = 10

Now we can plug these values into the formula

V = (1/3)10(729 + 225 + sqrt(729*225))

V = (1/3)10(954 + sqrt(164025))

V = (1/3)10(954 + 405)

V = (1/3)10(1359)

V = 4530 cubic units

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Step-by-step explanation:

(a) Let the original price of each 6-day ski pass be x. Then, the sale price of each pass is 0.85x = 272. Solving for x, we have:

0.85x = 272

x = 320

Therefore, the original price of each 6-day ski pass was $320.

(b) Two 3-day ski passes cost 2 x £110 = £220. Converting to US dollars using the exchange rate, we have:

£220 x 1.70 = $374

Therefore, Andrew would save:

$374 - $272 = $102

by buying one 6-day ski pass in the sale rather than two 3-day ski passes.

Answer:

Step-by-step explanation:

Для решения задачи мы можем использовать свойства треугольников и окружностей.

Первое, что мы можем заметить, это что треугольник ABD является равносторонним, так как все его стороны имеют одинаковую длину 13. Это означает, что угол ABD также равен 60 градусам.

Также мы можем заметить, что точка O является центром окружности, вписанной в треугольник ABD, так как все ее стороны касаются окружности в точке D. Из свойств вписанных углов, мы знаем, что угол AOD равен половине угла ABD, то есть 30 градусам.

Далее, мы можем заметить, что треугольник AOC является равнобедренным, так как угол ACO равен углу OCA (они оба равны 75 - 30 = 45 градусов), а сторона AC имеет одинаковую длину с стороной AB.

Таким образом, мы можем найти длину стороны AC, используя теорему косинусов для треугольника AOC:

AC^2 = AO^2 + OC^2 - 2 * AO * OC * cos(45)

Заметим, что AO = DO, так как точка O является центром вписанной окружности, а DO является радиусом этой окружности. Из прямоугольного треугольника ADO мы можем выразить DO как DO = AD/2 = 6.5.

Также, мы можем выразить OC, используя равенство углов в треугольнике ACO (ACO и AOD являются вертикальными углами):

ACO = AOD = 30 градусов

Тогда, угол OCA равен 180 - 2 * 45 = 90 градусам, что означает, что треугольник OCA является прямоугольным, и мы можем использовать теорему Пифагора:

OC^2 + AC^2 = OA^2

OC^2 + AC^2 = DO^2

AC^2 = DO^2 - OC^2

Теперь мы можем подставить выражения для DO и OC, и получить:

AC^2 = 6.5^2 - (6.5/sqrt(2))^2

AC^2 = 42.25 - 22.5625

AC^2 = 19.6875

AC = sqrt(19.6875)

AC = 4.43 (с точностью до сотых)

Таким образом, длина стороны AC равна пр

Step-by-step explanation:

Set it up as a ratio:

14 is to (14 +6)   as   21 is to  ?

14/20 = 21/?

? = 21 * 20 / 14 = 30 units

Therefore, each portion will be (1-x)/4 pounds.

Pounds (lb) is a unit of measurement of weight or mass commonly used in the United States, United Kingdom, and other countries that have adopted the Imperial system of measurement. One pound is equal to 0.453592 kilograms (kg). The symbol for pound is "lb", which comes from the Latin word libra. In everyday use, pounds are often used to measure the weight of objects, people, and animals, as well as food and other goods sold by weight.

Given by the question.

If you have used x pounds of the 1-pound box of oatmeal, then the remaining amount is 1 - x pounds.

You then divide this remainder into 4 equal portions, which means each portion will be (1-x)/4 pounds.

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Answer:

Step-by-step explanation:

To find the value of Z for a 95% confidence level, we can use a standard normal distribution table or a calculator that has a built-in function for finding Z values.

Using a calculator, we can use the following steps:

Determine the level of confidence, which is 95%. This means that the probability of the true population proportion being within the confidence interval is 0.95.

Find the critical value of Z using a Z-table or calculator. For a 95% confidence level, the critical Z value is 1.96.

Calculate the sample proportion, which is the number of married couples in the sample with at least one partner having a doctorate degree divided by the total sample size:

p-hat = 41/320 = 0.128125

Calculate the standard error of the sample proportion, which is the square root of the product of the sample proportion and the complement of the sample proportion, divided by the sample size:

SE(p-hat) = sqrt((p-hat)(1 - p-hat)/n) = sqrt((0.128125)(1 - 0.128125)/320) = 0.0248 (rounded to four decimal places)

Calculate the margin of error, which is the product of the critical Z value and the standard error:

Margin of error = Z * SE(p-hat) = 1.96 * 0.0248 = 0.0486 (rounded to four decimal places)

Calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error to the sample proportion:

Lower bound = p-hat - margin of error = 0.128125 - 0.0486 = 0.0795 (rounded to four decimal places)

Upper bound = p-hat + margin of error = 0.128125 + 0.0486 = 0.1767 (rounded to four decimal places)

Therefore, the 95% confidence interval for the percentage of married couples in which at least one partner has a doctorate degree is (0.0795, 0.1767).

The intersection of two parallel lines; x = 10 metres.

Alternate angles are created when two parallel lines are intersected by a transversal. Have a look at the given illustration; the two parallel lines are EF and GH. When a transversal splits two parallel lines, the alternate angles are equal.

The alternate interior angles are equal because of the parallel lines characteristic.

Angle STQ (denoted as 2x+10) and angle RQS are hence equal. Angle QRP (shown as x+20) and angle RQS are likewise equal.

Setting these two angles equal to each other, we can get:

x+20 = 2x+10

Simplifying this equation, we get:

x = 10.

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The base of the triangle is decreasing at a rate of 36/19 cm/min when the altitude is 9.5 cm and the area is 94 sq cm.

We can differentiate composite functions using the chain rule in calculus. Any function that is created by fusing two or more other functions is referred to as a composite function. According to the chain rule, if y = f(g(x)), and f and g are both differentiable functions, then the derivative of y with respect to x is given by:

dy/dx = f'(g(x)) * g'(x)

Given that, the altitude is 9.5 cm and the area is 94 sq cm.

The area of the triangle is given as:

A = 1/2(b)(h)

Differentiating both sides with respect to time we have:

dA/dt = 1/2[(db/dt)(h) + b(dh/dt)

Now,

dA/dt = 2 and dh/dt = 2.

Substituting the values we have:

2 = (1/2) * (db/dt * 9.5 + b * 2)

4 = db/dt * 9.5 + 2b......(1)

Using the area of triangle we know that:

Area = (1/2) * b * h

94 = (1/2) * b * 9.5

b = 20

Substituting the value of b in equation 1 we have:

4 = db/dt * 9.5 + 40

db/dt = - 36/19 cm/min

Hence, the base of the triangle is decreasing at a rate of 36/19 cm/min when the altitude is 9.5 cm and the area is 94 sq cm.

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The distance they ran is [tex]b * t1^2 * (d - 35) + 15 * t1 = 13t1.[/tex]

The length οf the line segment cοnnecting the twο sites is used tο measure distance between them. The shοrtest line segment between a pοint and a line will have a length equal tο the distance between them.

With the data abοve, we can create twο equatiοns using the fοrmula distance = speed x time:

Equatiοn 1: (d - 15) = b * t1

Equatiοn 2: (d - 35) = c * t1

Where t1 is the amοunt οf time it tοοk Anne tο cοmplete the race.

Carl cοvered a distance οf d - 22 after Bοris had dοne his 22 m οf running. The same fοrmula can be used tο create a different equatiοn:

Equatiοn 3: (d - 22) = c * t2

where t2 is the amοunt οf time it tοοk Bοris tο cοmplete the race.

Equatiοns 1 and 2 can be used tο find b and c in terms οf t1:

b = (d - 15) / t1

c = (d - 35) / t1

Substituting these equatiοns intο Equatiοn 3, we get:

(d - 22) = (d - 35) / t1 * t2

Simplifying, we get:

t1 * t2 = 13 / c

In the previοus equatiοn, we may substitute the fοrmulas fοr b and c in terms οf t1 tο οbtain:

t1 * t2 = 13t1 / (d - 35)

Simplifying, we get:

t2 = 13 / (d - 35)

Tο sοlve fοr b in terms οf t1, we can insert the fοllοwing fοrmula fοr t2 intο Equatiοn 1:

(d - 15) = b * t1

(d - 15) = ((d - 35) / t1) * 13 / (d - 35) * t1

Simplifying, we get:

b = 13 / t1 - 2

Substituting this expressiοn fοr b intο Equatiοn 2, we get:

(d - 35) = c * t1

(d - 35) = ((d - 15) / t1 - 13 / t1 + 2) * t1

Simplifying, we get:

c = (d - 35) / t1 + 13 / t1 - 2

Nοw we can create anοther equatiοn using these expressiοns fοr b and c in terms οf t1:

d = (d - 15) + 15 = b * t1 + 15

d = (d - 22) + 22 = c * t2 + 22 = (d - 35) / t1 * 13 / (d - 35) + 22

When we equalize these twο expressiοns, we οbtain:

b * t1 + 15 = (d - 35) / t1 * 13 / (d - 35) + 22

Multiplying bοth sides by t1 * (d - 35), we get:

b * t1² * (d - 35) + 15 * t1 * (d - 35) = 13t1² + 22t1 * (d - 35)

Simplifying, we get:

b * t1² * (d - 35) + 15 * t1 = 13t1

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a) The probability that a randomly selected subprime mortgage in the state during November 2008 was neither in foreclosure nor repossessed is 0.50.

b) The probability that a randomly selected subprime mortgage in the state during November 2008 was not current is 0.32.

Probability describes the chance or likelihood that an expected outcome or event does or does not occur.

Probability is the quotient of the expected outcome over the total number of possible outcomes or events.

Probabilities can be stated in ratios (fractions, decimals, or percentages).

Mortgage Status

                  Current    Past Due    In Foreclosure    Repossessed    Total

Number     156,330      58,310              8,550                5,590       228,780

The number of subprime mortgage neither in foreclosure nor repossessed = 114,640 (156,330 + 58,310)

The probability that a subprime mortgage in the state was neither in foreclosure nor repossessed = 0.50 (114,640/228,780).

The number of subprime mortgage that was not current = 72,450 (58,310 + 8,550 + 5,590) or (228,780 - 156,330)

The probability that a subprime mortgage in the state was not current = 0.32 (72,450/228,780).

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The slope of the equation [tex]y = -x + 3 is -1[/tex]  , and its y-intercept is 3. In addition, the line  [tex]y = -x + 3[/tex] has 3 as its x-intercept.

Determine the slope of the line by determining the rise and the run using two of the line's points.

The phrases "rise" and "run" are used to indicate height differences between two sites. for a - a - a - the - the - the - the Run is equal to rise plus slope. Slope is the result of adding rise and run.

The sum of cubes problem is the solution to the equation  [tex]x3+y3+z3=k.[/tex] Although the equation appears simple.

It becomes exponentially more challenging to answer when it is phrased as a "Diophantine equation" the values for x, y, and z must all be whole numbers for any value of k in the issue.

Therefore, The equation  [tex]y = -x + 3[/tex] has a y-intercept of 3, and its slope is -1. In addition, 3 is the x-intercept of the line [tex]y = -x + 3.[/tex]

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Answer:

Step-by-step explanation:

y2-y1/x2-x1

-7-(-19)/-2-1

12/-2

-6

The slope is -6

After answering the provided question, we can conclude that slant height  of pyramid  [tex]= \sqrt((2/2)^2 + 5^2) = \sqrt(29) = 5.39 units.[/tex]

A pyramid is a polygon formed by connecting points known as bases and polygonal vertices. For each hace and vertex, a triangle known as a face is formed. A cone with a polygonal shape. A pyramid with a floor and n pyramids has n+1 vertices, n+1 vertices, and 2n edges. Every pyramid is dual in nature. A pyramid contains three dimensions. A pyramid is made up of a flat tri face and a polygonal base that come together at a single point known as the vertex. A pyramid is formed by connecting the base and peak. The edges of the base form triangle faces known as sides, which connect to the top.

            /\

          /    \

        /         \

      /______\

         5

         |

         |

         |

         |

         |

         2

The square pyramid in the diagram above has a two-unit-long square base and four five-unit-high triangular faces. The Pythagorean theorem can be used to calculate the slant height of each triangular face:

slant height [tex]= \sqrt((2/2)^2 + 5^2) = \sqrt(29) = 5.39 units.[/tex]

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FALSE. Every random sample of the same size from a given population will not produce exactly the same confidence interval for μ.

The confidence interval is a statistical measure used to estimate the range of values within which a population parameter is likely to fall. The confidence interval is calculated based on the sample mean and standard deviation, as well as the level of confidence desired.

Suppose we take a random sample of size n from a population, and calculate the confidence interval for the population mean using this sample. The sample mean and the sample standard deviation will be used to estimate the true population mean and the population standard deviation, respectively. However, as the sample is random, each sample—despite being drawn from the same population—will have different values for the sample mean and standard deviation. Thus, different samples will produce different confidence intervals for the population mean.

Moreover, the size of the sample also affects the width of the confidence interval; larger samples tend to produce more precise estimates of the population mean, while smaller samples yield larger confidence intervals. Therefore, random samples of different sizes from a given population will also produce different confidence intervals.

In summary, the confidence interval is a statistical measure that provides a range of likely values for the population parameter, such as the population mean. While it can be calculated using any random sample from a population, different samples of the same size or different sizes will generally produce different confidence intervals.

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Answers:

a) geometric

b) arithmetic

c) arithmetic

d) geometric

Explanations:

An arithmetic sequence is a when the next term is achieved by adding/subtracting by a constant (the same number) to the current term.

A geometric sequence is a when the next term is achieved by multiplying/diving the current term by a constant. (i.e. the ratio between any two consecutive terms is the same for any other two consecutive terms).

a) The numbers are all products of the previous term by a multiple of 2 (i.e. each term is multiplied by 2 to get the next term).

b) The numbers increase by the same amount between every term (by 3 in this case).

c) The numbers increase by the same amount between every term (by 5 in this case).

d) The numbers are all products of the previous term by a multiple of 10 (i.e. each term is multiplied by 10 to get the next term).

R(x) = 47x denotes the total revenue function, and C(x) = 100 + 30x + 0.1x2 is the total cost function.

(a) P(100) = -200

(b) MP(x) = 47 - (30 + 0.2x)

(c) MP(100) = -3

(d) P(101) - P(100) = -0.2

(a) P(100) represents the profit made when x=100 units are sold. It can be calculated as follows:

P(x) = R(x) - C(x)

P(100) = R(100) - C(100)

P(100) = [tex]47(100) - (100 + 30(100) + 0.1(100)^2)[/tex]

P(100) = 4700 - 4000 - 100

P(100) = -200

(b) The marginal profit function MP represents the rate of change of profit with respect to the number of units sold. It can be calculated as follows:

MP(x) = R'(x) - C'(x)

MP(x) = 47 - (30 + 0.2x)

(c) MP(100) represents the marginal profit at x=100 units. It can be calculated by substituting x=100 into the marginal profit function:

MP(100) = 47 - (30 + 0.2(100))

MP(100) = 47 - 50

MP(100) = -3

Profit will drop by |MP(100)| dollars at x = 100, according to MP(100).

(d) P(101) - P(100) represents the additional profit made when the 101st unit is sold compared to when the 100th unit is sold. It can be calculated as follows:

P(101) - P(100) = R(101) - C(101) - R(100) + C(100)

P(101) - P(100) =[tex]47(101) - (100 + 30(101) + 0.1(101)^2) - 47(100) + (100 + 30(100) + 0.1(100)^2)[/tex]

P(101) - P(100) = 47 - 30 - 0.2(101) - 47 + 30 + 0.1(100)

P(101) - P(100) = -0.2

The sale of the 101st unit will increase profit by $0.20.

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Answer:

According to the scale, 1 inch on the drawing represents 1.5 feet in real life. So, to find the actual length of the door, we need to multiply the length on the drawing by the scale factor:

4 inches x 1.5 feet/inch = 6 feet

Similarly, to find the actual width of the door, we need to multiply the width on the drawing by the scale factor:

7 inches x 1.5 feet/inch = 10.5 feet

Therefore, the actual measurements of the door are 6 feet by 10.5 feet.

Answer:

If the graph of f (x) = 3 is reflected over the x-axis, the equation of the new graph is g(x) = -3, so the correct answer is D. g(x) = -().

The value of the following decimal floating-point number as a single precision floating point values following the ire std 754-1985111.125 is

1100    0001     0110     0100    0000     0000     0000    0000

= C1640000H

IEEE-754 representation for float (single-precision) type is as follows

Sign - 0

Exponent - 1 to 18

Mantissa - 9 to 31

So the exponent field is 8 bits and the mantissa is 23 bits (the precision is actually 24 bits due to the implicit 1 needed in the normalized representation; IEEE 754 also allows demoralized numbers close to 0, but this does not does not apply to the particular question).

Exponential fields also require signs to represent fractions. IEEE 754 does this by giving a bias - 127 for single precision, which means we just subtract 127 from the suggested value to get the actual value. So 0 becomes -127 and 255 (the maximum 8-bit value becomes 128.

Now, coming to the given question we need to represent -14.25 which equals -1110.01 in binary.

Converting to normalized form (only one to the left ) we get

-11101.01 =-1.11001 × 2³

Since we omit the implied  in IEEE-754 representation we get

Mantissa bits = 11001

Exponent bits = 11 + 0111111 = 10000010 (Adding bias Sign bit 1)

(since number is negative)

This will be 1 × 110000010 = 11001000000000

Grouping in 4 bits to convert ot Hexadecimal we get

1100    0001     0110     0100    0000     0000     0000    0000

= C1640000H

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Answer:

B is correct

Step-by-step explanation:

If it is spun twice then the probability of it landing on 2 and then an odd number is:

Pr(2,1) or Pr(2,3) or Pr(2,5) or Pr(2,7)

1/49 * 4

4/49

The correct option b) [tex]\rm y = \sqrt{x+ 3} + 3[/tex] is a function of the given graph.

The graph οf a relatiοn is said tο reflect a functiοn if a vertical line drawn anywhere οn the graph οnly intersects the graph οnce. In the event when a vertical line can graph is nοt a representatiοn οf a functiοn if there are mοre than twο pοints οn it.

The graphs coordinate is at (-3, 3)

Lets put the value in all the given function to find the correct function

a. [tex]\rm y = \sqrt{x - 3} + 3[/tex]

Using (-3, 3) as x and y

[tex]\rm 3 = \sqrt{(-3) - 3} + 3[/tex]

[tex]\rm 3 = \sqrt{9} + 3[/tex]

[tex]\rm 3 \neq 3 + 3[/tex]

[tex]\rm y = \sqrt{x - 3} + 3[/tex] is not a function of graph.

b. [tex]\rm y = \sqrt{x+ 3} + 3[/tex]

Using (-3, 3) as x and y

[tex]\rm 3 = \sqrt{-3+ 3} + 3[/tex]

[tex]\rm 3 = \sqrt{0} + 3[/tex]

[tex]\rm 3 = 0 + 3[/tex]

[tex]\rm 3 = 3[/tex]

[tex]\rm y = \sqrt{x+ 3} + 3[/tex] is a function of the given graph.

The other two options are no the functions as cube root is used and the value of x wont satisfy there.

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Answer:

Let x be the amount invested in Treasury bonds and y be the amount invested in mutual funds. Then we have the following constraints:

x + y ≤ 40 (total investment cannot exceed $40 million) x ≥ 5 (at least $5 million must be invested in Treasury bonds) y ≥ 15 (at least $15 million must be invested in mutual funds) 100x + 200y ≤ 7000 (total fees cannot exceed $7000)

The objective is to maximize the annual interest, which is given by:

0.06x + 0.08y

We can solve this problem using linear programming. The feasible region is a polygon with vertices at (5, 15), (5, 35), (30, 10), and (40, 0). We evaluate the objective function at each vertex:

At (5, 15): 0.06(5) + 0.08(15) = 1.2At (5, 35): 0.06(5) + 0.08(35) = 2.6At (30, 10): 0.06(30) + 0.08(10) = 2.8At (40, 0): 0.06(40) + 0.08(0) = 2.4

The maximum value of the objective function is 2.8, which occurs at (30, 10). Therefore, the amount that should be invested in Treasury bonds is $30 million and the amount that should be invested in mutual funds is $10 million.

The annual interest earned is 0.06(30) + 0.08(10) = $2.8 million.

Step-by-step explanation:

Answer:

Perceptions of same-sex marriage: In June 2016, a CBS News poll asked a sample of adults worldwide whether it should be legal or not legal for same-sex couples to marry (reported at http://www.pollingreport.com). The opinions of adults worldwide were as follows: 58%, legal; 33%, not legal; and 9%, unsure/no answer.

What type of distribution is this? __________________________

Knowing that 1,280 adults were polled nationwide, how many Americans polled felt that same-sex couples should be allowed to legally marry? __________________________

Step-by-step explanation:

The distribution of opinions on the legality of same-sex marriage worldwide is not described in the prompt, so it is not possible to determine the type of distribution. However, it is possible to use the proportions provided to estimate the number of Americans who felt that same-sex couples should be allowed to legally marry.

If 58% of all adults worldwide believe that same-sex marriage should be legal, and 1,280 adults were polled nationwide, we can estimate the number of Americans who hold this opinion as:

0.58 x 1,280 = 742.4

Therefore, an estimated 742 Americans polled felt that same-sex couples should be allowed to legally marry.

Answer:

Step-by-step explanation:

We can factor out a common factor of -4x from the equation:

y = -4x(x² + 4x - 21)

To factor the quadratic expression in the parentheses, we need to find two numbers that multiply to -21 and add to 4. These numbers are 7 and -3:

y = -4x(x + 7)(x - 3)

To check our work, we can multiply the three factors:

y = -4x(x + 7)(x - 3) = -4x(x² + 4x - 21) = -4x³ - 16x² + 84x

So the factored form is y = -4x(x + 7)(x - 3), and the check shows that we have factored the equation correctly.

By analyzing the relationship between the distance traveled and the time taken from the given graph, we determined that the cruising speed of the airplane is 567 units of distance per 1 unit of time.

To find the cruising speed of the airplane, we can analyze the given graph and observe the relationship between the distance traveled and the time taken.

Looking at the data points provided, we see that the distance traveled is directly proportional to the time taken. This means that for every unit increase in time, there is a corresponding increase in the distance traveled.

Let's consider the first two data points: (1,567) and (2,1134). The time difference between these two points is 2 - 1 = 1, and the distance traveled difference is 1134 - 567 = 567.

Since the distance traveled is directly proportional to time, we can conclude that the speed of the airplane is equal to the distance traveled divided by the time taken. In this case, the speed would be 567 units of distance per 1 unit of time.

Now, let's calculate the cruising speed using the other data points as well:

(2,1134) - (1,567) = 1134 - 567 = 567 units of distance per 1 unit of time

(3,1701) - (2,1134) = 1701 - 1134 = 567 units of distance per 1 unit of time

(4,2268) - (3,1701) = 2268 - 1701 = 567 units of distance per 1 unit of time

(5,2268) - (4,2268) = 2268 - 2268 = 0 units of distance per 1 unit of time

We can see that for each time interval, the distance traveled is always 567 units. Therefore, we can conclude that the cruising speed of the airplane is 567 units of distance per 1 unit of time.

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If x is an integer, the least possible value of x is 4. So the option D is correct.

The triangle's third side should be less than the sum of the other two sides and more than the difference of the other two sides.

11 - 7.5 < x < 11 + 7.5

Simplify

3.5 < x < 18.5

So the value of the x is between 3.5 and 18.5.

From the option the value 4 and 5 lies between 3.5 and 18.5.

As we have to determine the least possible value of x, so the value of x should be 4. So the option D is correct.

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The complete question is:

The sides of a triangle have lengths 7.5, 11, and x. If x is an integer, what is the least possible value of x?

A. 1

B. 2

C. 3

D. 4

E. 5

the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

How to find?

The given function is:

f(x) = 1 / (x² + 10x + 25)

To find the definite integral of this function over the interval [5, 7], we can use the following steps:

Rewrite the function using partial fraction decomposition:

f(x) = 1 / (x² + 10x + 25)

= 1 / [(x + 5)²]

Using partial fraction decomposition, we can write this as:

f(x) = A / (x + 5) + B / (x + 5)²

where A and B are constants to be determined. Multiplying both sides by the common denominator (x + 5)², we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

1 = 5A + B

= 5A + 1

Solving for A, we get:

A = 0

Therefore, the partial fraction decomposition is:

f(x) = 1 / [(x + 5)²]

= 0 / (x + 5) + 1 / (x + 5)²

Use the formula for the definite integral of a power function:

∫ xⁿ dx = (1 / (n + 1))× x²(n + 1) + C

where C is the constant of integration.

Using this formula, we can find the antiderivative of the function 1 / (x + 5)²:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

Evaluate the definite integral over the interval [5, 7]:

∫[5,7] 1 / (x + 5)² dx

= [-1 / (x + 5)] [from 5 to 7]

= (-1 / 12) - (-1 / 10)

= (-5 / 600)

Therefore, the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

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Yes, you can stop checking for factor pairs when you find a pair that repeats.

A factor pair is defined in mathematics as a set of two factors that, when multiplied together, produce a specific product. In other words, it is a set of two numbers that we multiply to get a product. For example, in the multiplication statement, 6 7 = 42, 6 and 7 is one of the factor pairs that gives us the product 42.

This is due to the fact that each number has a unique set of factors. When you find a factor pair that repeats, you know you've found all of the factors of that number. Any additional pairs you find will simply be a permutation of the same factors you've already discovered.

Consider the number 24 as an example. Its components are as follows:

1, 2, 3, 4, 6, 8, 12, 24

When looking for factor pairs, we begin with 1 and 24, then move on to 2 and 12, 3 and 8, and finally 4 and 6. We now have all of the factors of 24 because we discovered a pair (4, 6) that repeats the same factors as an earlier pair (6, 4).

As a result, once you find a pair of factors that repeats, you can be confident that you have discovered all of the factors of the number you are investigating.

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