Given the points (2,k) and (0,−6), for which values of k would the distance between the points be 5–√ ?

Answer:

Option A

Step-by-step explanation:

   [tex]\sf Slope=\dfrac{Change \ in \ y}{Change \ in \ x}=\dfrac{rise}{run}[/tex]

We can find the equation of line, by comparing the slope of line segment AC and slope of AE.

Slope of AC = Slope of AE

           [tex]\sf \dfrac{4}{3}=\dfrac{y +2}{x}[/tex]

Cross multiply

4*x = 3*(y +2)

4x   = 3y + 3*2

4x  = 3y + 6

Equation of the line in  standard form:

         4x - 3y - 6= 0

The orthocentre for the triangle ABC is (1.85, 5.88).

Triangle ABC has vertices at A(2, 8), B(6, 2) and C(-3, 2).

We need to use analytical geometry to determine the coordinates of the orthocentre.

Steps to find the orthocentre:

Step 1: First, we will find the slopes of any two sides of the triangle (say AC and AB).

Step 2: Next, we can find the slopes of the corresponding altitudes. Remember that if two lines are perpendicular to each other, they satisfy the following equation.

Step 3: Next, we will use the slope-point form of the equation of a straight line to find the equations of the lines that are coincident with the altitudes BE and AD.

Step 4: Next, we can solve the equations of BE and CF simultaneously to find their solution, which gives us the coordinates of the orthocentre H.

Now, the slope of AC=2-8/-3-2=6/5 and the slope of BE=5/6.

The slope of AB=2-8/6-2=-6/4=-3/2 and the slope of CF=-2/3

The slope of BE=y-2/x-6=5/6

⇒6x-5y-26=0----(1)

The slope of CF=y-2/x+3=-2/3

⇒-2x-3y=0----(2)

By solving (1) and(2), we get

y=26/14=13/7=1.85 and x=5.88

Therefore, the orthocentre for the triangle ABC is (1.85, 5.88).

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The sample size required is 1725.

(A) Sample size calculation

p=0.15

For 98%, z=2.326

d=0.02

Sample size = (z^2*p*(1-p))/d^2

= (2.326^2*0.15*0.85)/0.02^2

=1724.5

The sample size required= 1725

(B) Sample size calculation

p=0.5

For 98%, z=2.326

d=0.02

Sample size = (z^2*p*(1-p))/d^2

= (2.326^2*0.5*0.5)/0.02^2

= 3381.4

The sample size required= 3382

In records, the pattern size is the degree of the variety of character samples utilized in a test. For example, if we're testing 50 samples of folks that watch television in a town, then the pattern length is 50.

Sample size willpower is the act of selecting the range of observations or replicates to consist of in a statistical pattern. The sample size is a crucial function of any empirical look at in which the intention is to make inferences about a populace from a sample.

Your question is incomplete. Please find the complete question below.

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with98% confidence if

(a) Does he use a 2006 estimate of 15% obtained from a Coinstar National Currency Poll?

(b) he does not use any prior estimate?

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Question:

A teacher puts students' names in a hat and chooses without looking to get a sample of 3 students. What type of sample is this?

Answer:

Simple random

[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]

[tex]\qquad \tt \rightarrow \:x = -3 [/tex]

____________________________________

[tex] \large \tt Solution \: : [/tex]

[tex]\qquad \tt \rightarrow \: CD + DE = CE[/tex]

[tex]\qquad \tt \rightarrow \: x + 10 + x + 4 = 8[/tex]

[tex]\qquad \tt \rightarrow \: 2x + 14 = 8[/tex]

[tex]\qquad \tt \rightarrow \: 2x =8 - 14[/tex]

[tex]\qquad \tt \rightarrow \: 2x = - 6[/tex]

[tex]\qquad \tt \rightarrow \: x = - 3[/tex]

Answered by : ❝ AǫᴜᴀWɪᴢ ❞

[tex]\huge \boxed{\sf x=-3}\\\\\\\displaystyle \sf CD+DE=8 \\\\x+10+x+4=8\\\\2x+14=8\\\\Subtracting\ 14\ from\ each\ side\\\\ 2x+14-14=8-14\\\\2x=-6\\\\ Dividing\ each\ side\ by\ 2 \\\\ \frac{2x}{2} =\frac{-6}{2} \\\\x=-3[/tex]

Look at the multiples of lcm(3, 8) = 24, each of which are divisible by both 3 and 8. Naturally, the number we're looking for must have at least two digits.

The last/ones digit of a number [tex]n[/tex] can be computed by evaluating [tex]n \pmod{10}[/tex]. Suppose the ones digit is 1. Then

[tex]n \equiv 24k \equiv 20k + 4k  \equiv 4k \equiv 1 \pmod{10}[/tex]

but this has no solution since no multiple of 4 ends in 1. So our number ends with a 2.

If our number has two digits, then in order for it to be divisible by 3, the first digit must be 1, since 1 + 2 = 3. But 12 is not divisible by 24.

If our number has three digits, then it's either 111 (with digital sum 1 + 1 + 1 = 3) or 222 (digital sum 2 + 2 + 2 = 6), since no other triplet of 1s and 2s sums to a multiple of 3. But 111 = 3×37 and 222 = 2×111, so neither are divisible by 24.

Now suppose [tex]n[/tex] has four digits. Also suppose that the tens digit is 2. Then our number must be 1122 to uphold divisibility by 3, but 1122 = 2×3×11×17 is not divisible by 8. So the tens digit must be 1. This in turn leaves two remaining candidates, 1212 and 2112. We have 1212 = 12×(100 + 1) not divisible by 8, so our number must be [tex]\boxed{n=2112}[/tex]. And it is, since 2112 = 24×88.

Answer:

12 and 14

Step-by-step explanation:

The first number can be assigned x, the second will be assigned x+2 so that it will be even.

3(x)+5(x+2) = 106

3x+5x+10 = 106

8x = 96

x = 12

x+2 = 14

The two consecutive even numbers such that the sum of three times the smaller and five times the larger is 106 are 12 and 14, computed using the linear equation in one variable 3x + 5(x + 2) = 106.

We are given two consecutive even numbers, thus the larger number will be two more than the smaller one, as consecutive even numbers differ by two.

We assume the smaller number to be x.

Thus, the larger number = x + 2.

We have been given that three times the smaller number summed with five times the larger number results in 106.

This can be shown by the linear equation in one variable:

3x + 5(x + 2) = 106.

To find the numbers, we need to solve this linear equation in one variable as follows:

3x + 5(x + 2) = 106,

or 3x + 5x + 10 = 106 {Simplifying},

or, 8x + 10 = 106 {Simplifying},

or, 8x + 10 - 10 = 106 - 10 {Subtracting 10 from both sides of the equation},

or, 8x = 96 {Simplifying},

or, 8x/8 = 96/8 {Dividing both sides of the equation by 8},

or, x = 12 {Simplifying}.

Thus the smaller number, x = 12.

The larger number, x + 2 = 12 + 2 = 14.

Thus, the two consecutive even numbers such that the sum of three times the smaller and five times the larger is 106 are 12 and 14, computed using the linear equation in one variable 3x + 5(x + 2) = 106.

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Answer:

-14

Step-by-step explanation:

Given -x+5y=7

Now we substitute x+5y into -2(-x+5y)

-2(-x+5y) = (-2)(7)

= - 14

The ages of Lute,  Maya and Aran are respectively; 24, 8 and 6 years old.

Let Lute's age be x

Thus;

Maya's age = ¹/₃x

Aran's age = ¹/₄x

We are told the sum of their ages is 38. Thus;

x + ¹/₃x + ¹/₄x = 38

Multiply through by 12 to get;

12x + 4x + 3x = 456

19x = 456

x = 456/19

x = 24

Maya's age =  ¹/₃ * 24 = 8 years

Aran's age = ¹/₄ * 24 = 6 years

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Answer:

36 total students in the class

Step-by-step explanation:

There are 24 boys in the class.

Since they sit in pairs, and exactly 1/4 of the boys sit with a girl, then one-quarter of the 24 boys sits with a girl.

1/4 * 24 = 6, so 6 of the boys sit with a girl.  Reflexively, 6 girls sit with a boy.

Since the problem states that exactly half of the girls sit with a boy, then half of the total number of girls is 6 (since we know that exactly 6 girls were sitting with a boy).

1/2 * x = 6

Multiplying both sides by 2...

x=12

So there are 12 girls total in the class, half (6) of whom sit with a boy.  

There are 24 boys total in the class, one-quarter (6) of whom sit with a girl.  The rest of the boys (18) sit in pairs with another boy (9 pairs), and the rest of the girls (6) sit with another girl (3 pairs).

Since there are 12 girls total and 24 boys total in the class, the total number of students is the sum of 12 and 24

12+24 =36 total students in the class

Answer:

45units^2

Step-by-step explanation:

Break the polygon into two shapes. A rectangle and a triangle.

The area formula for a rectangle is A=L•W or

Area = Length x Width. Counting the units for length and width is 9 x 2. Which would equal 18units.

Now for the triangle, the formula is A=1/2 (b•h)

The base is 9 and Height is 6.

9 • 6 = 54

Since a triangle is half the area of a rectangle you divide 54 by 2. Area for triangle is 27.

Now add the rectangle area and triangle area.

18+27=45

Area=45 units ^2

Answer: 6.8

Step-by-step explanation:

By the Law of Cosines,

[tex]c^2 = 5^2 + 9^2 - 2(5)(9)\cos 48^{\circ}\\\\c=\sqrt{5^2 + 9^2 - 2(5)(9)\cos 48^{\circ}}\\\\c \approx 6.8[/tex]

Answer:

4(2x+7) which is equal to 8x+28

A number in a fraction implies the part of a number in another given number. While percentage relates a given number to 100%. Thus the answers to the given question are:

a. a fraction of penalty shots that the team missed = [tex]\frac{81}{100}[/tex]

b. percentage of penalty shots missed =  [tex]\frac{81}{100}[/tex] x 100%

                                                           = 81%

A fraction is a part of a given number stated in a quotient form. Generally, all numbers can be expressed in form of fractions, even whole numbers.

The percentage is an expression that shows the part of a number in a given number with respect to 100%. An example is 10%([tex]\frac{10}{100}[/tex]).

From the given question, let;

n, number of penalty shots made = 19

T, the total number of penalties taken = 100

Thus, the number of penalties missed = T - n

                                               = 100 - 19

                                               = 81

Thus,

i. a fraction of penalty shots that the team missed = [tex]\frac{81}{100}[/tex]

ii. percentage of penalty shots missed =  [tex]\frac{81}{100}[/tex] x 100%

                                                           = 81%

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Answer:

Step-by-step explanation:

Discussion.

Not directly. But the quadratic formula can do it. But that's not your question.

the factors you get must contain the factors for 1 which are 1 and - 1

These factors must add to - 5. There's no way that will happen with 1 and - 1 and you would be creative math if you tried to say that you could make one of the factors (x -1-1-1-1-1-1). That creates a whole new question.

[tex]in \: pounds \\ 12 \times 288 = 3456 \: pounds[/tex]

[tex]1 \: kilo = 2.2046 \: pounds[/tex]

[tex]x \: kilos = 3456 \: pounds[/tex]

[tex]x = \frac{3456}{2.2046} = 1567.61 \: kilos[/tex]

Answer:

Angle Q is acute.

Step-by-step explanation:

In an obtuse triangle, there is one obtuse angle and the other two angles are acute.

⊰________________________________________________________⊱

Answer:

x=5

Step-by-step explanation:

Apply the distributive property,

2(x+6)-10=12

2x+12-10=12

Collect a few like terms

2x+2=12

Subtract 2 from both sides

2x=10

Divide the equation by 2.

x=5

Done!!

⊱______________________________________________________⊰

2(x+6) - 10 = 12
2(x+6) = 22
x+6 = 11
x = 11 - 6 = 5

x = 5

The system of inequalities is:

v + h ≤ 8

v < 2

h ≥ 1.5

Let's define the variables:

The maximum time that you can spend on both activities is 8 hours, then:

v + h ≤ 8

You want to spend less than 2 hours on video games, so:

v < 2

You want to spend at least, 1.5 hours on homework, so:

h ≥ 1.5

Then the system of inequalities is:

v + h ≤ 8

v < 2

h ≥ 1.5

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Answer:

SSS similarity

Step-by-step explanation:

From the smaller triangle to the larger triangle, the ratio of each pair of corresponding sides is 2.

Answer:

Step-by-step explanation:

Answer:

[tex]\frac{\sqrt{61} }{\sqrt{97} }[/tex]

Step-by-step explanation:

Sin(y) = opposite over hypotenuse

opposite side is [tex]\sqrt{61}[/tex]

hypotenuse is [tex]\sqrt{97}[/tex]

It will take 29.4 years for the population of Africa to be two times what it is today. This reflects: doubling time.

Doubling time is the time it takes for a particular amount of size or value to double at a given growth rate. By applying the 70 rules, we can find the doubling time of the exponentially increasing population. To do this, divide 70 by the growth rate (r).

Rule of 70 is a simplified way to determine the doubling time using the equation doubling time = 70 / r. Where r is the percentage growth rate of the population. For example, if the population of 10 species increases by 2 individuals per year, the r value will be 20%.

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The probability of getting 3 or more who were involved in a car accident last year is 0.126.

Given 9% of the drivers were involved in a car accident last year.

We have to find the probability of getting 3 or more who were involved in a car accident last year if 14 were selected randomly.

We have to use binomial theorem which is as under:

n[tex]C_{r}p^{r} (1-p)^{n-r}[/tex]

where p is the probability an r is the number of trials.

Probability that 3 or more involved in a car accident last year if 14 are randomly selected=1-[P(X=0)+P(X=1)+P(X=2)]

=1-{[tex]14C_{0}0.9^{0} (0.1)^{14} +14C_{1} (0.9)^{1} (0.1)^{13} +14C_{2} (0.9)^{2} (0.1)^{12}[/tex]}

=1-{1*0.2670+14!/13!*0.9*0.29+14!/2!12!*0.0081*0.2358}

=1-{0.2670+0.3654+0.2358}

=1-0.8682

=0.1318

Among the options given the nearest is 0.126.

Hence the probability that 3 or more are involved in the accident is 0.126.

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[tex]\rm \dfrac{ x^2 }{100} + \dfrac{ y^2}{36} = 1[/tex] is the equation of the ellipse.

The equation of an ellipse is given by

[tex]\rm \dfrac{ (x-h)^2 }{a^2} + \dfrac{ (y-k)^2}{b^2} = 1[/tex]

here

(h,k) is the center

c = is the distance between center to focus

c = [tex]\rm \sqrt{a ^2 - b^2}[/tex]

as the a = 10 , component length

and c = 8

8 = [tex]\rm \sqrt{10 ^2 - b^2}[/tex]

8² = 10² -b²

b² = 36

b = 6

h = 0 ,k = 0

Substituting the values given

[tex]\rm \dfrac{ (x-0)^2 }{10^2} + \dfrac{ (y-0)^2}{6^2} = 1[/tex]

[tex]\rm \dfrac{ x^2 }{100} + \dfrac{ y^2}{36} = 1[/tex]

Therefore the equation of the ellipse has been determined.

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Answer:

Its 31.0 as the the two triangles are similar and the scale factor is 2

The daughter's age in 6 years time is 20

Let their current age be x and y

So, we have:

x = 3y

7 years ago, we have:

x - 7 : y - 7 = 5 : 1

Express as fraction

[tex]\frac{x - 7}{y - 7} = \frac 51[/tex]

Cross multiply

x - 7 = 5y - 35

This gives

x = 5y - 28

Substitute x = 3y in x = 5y - 28

3y = 5y - 28

Evaluate the like terms

2y = 28

Divide by 2

y = 14

In 6 years time, we have:

y + 6 = 20

Hence, the daughter's age in 6 years time is 20

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The figures have corresponding sides that are proportional, therefore, they are similar figures.

When two figures are similar to each other, the corresponding sides are proportional to each other.

The two figures given have two opposite sides that are parallel and congruent to each other. Therefore:

DA/VS = 21/7 = 3

AB/ST = 15/5 = 3

Thus, we can conclude that since their corresponding sides are proportional, therefore, they are similar to each other.

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The length of extra cable that is required to connect the two pieces of existing cable is equal to 182 meters.

In order to determine the length of extra cable that is required to connect the two pieces of existing cable, we would apply the law of cosine as follows:

B² = A² + C² - 2(A)(C)cosB

Substituting the given parameters into the formula, we have;

B² = 325.0² + 430.0² - 2(325.0)(430.0)cos23

B² = 105,625 + 184,900 - 279,500(0.9205)

B² = 290,525 - 257,279.75

B² = 33,245.25

B = √33,245.25

B = 182.33 ≈ 182 meters.

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