Raina is putting 13 colored light bulbs into a string of lights. There are 5 white light bulbs, 6 orange light bulbs, and 2 blue light bulbs. How many distinct orders of light bulbs are there if two light bulbs of the same color are considered identical (not distinct)?

Answer: confidence interval = 27.5 +/- 1.68

= ( 25.82, 29.18)

Step-by-step explanation:

Given;

Number of samples n = 64

Standard deviation r = 6mi/gallon

Mean x = 27.5mi/gallon

Confidence interval of 97.5%

Z' = t(0.0125) = 2.24

Confidence interval = x +/- Z'(r/√n)

= 27.5 +/- 2.24(6/√64)

= 27.5 +/- 1.68

= ( 25.82, 29.18)

Answer:

[tex]p_v =2*P(Z>4.21) =2.55x10^{-5}[/tex]

And we can use the following excel code to find it:"=2*(1-NORM.DIST(4.21,0,1,TRUE)) "

With the p value obtained and using the significance level assumed for example[tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the percentage 1 is significantly different from the percentage 2.

D) Are statistically different.

Step-by-step explanation:

The system of hypothesis on this case are:  

Null hypothesis: [tex]\mu_1 = \mu_2[/tex]  

Alternative hypothesis: [tex]\mu_1 \neq \mu_2[/tex]  

Or equivalently:  

Null hypothesis: [tex]\mu_1 - \mu_2 = 0[/tex]  

Alternative hypothesis: [tex]\mu_1 -\mu_2\neq 0[/tex]  

Where [tex]\mu_1[/tex] and [tex]\mu_2[/tex] represent the percentages that we want to test on this case.

The statistic calculated is on this case was Z=4.21. Since we are conducting a two tailed test the p value can be founded on this way.

[tex]p_v =2*P(Z>4.21) =2.55x10^{-5}[/tex]

And we can use the following excel code to find it:"=2*(1-NORM.DIST(4.21,0,1,TRUE)) "

With the p value obtained and using the significance level assumed for example[tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the percentage 1 is significantly different from the percentage 2.

And the best option on this case would be:

 D) Are statistically different.

Total envelopes: 3 + 1 + 3 + 2 = 9 with a total of 3 green ones.

Probability of picking green first: 3 out of 9 = 3/9 = 1/3

There are 8 envelopes left, with 2 green ones.

Probability of picking a green one is 2 out of 8 = 2/8 = 1/4

There are 7 envelopes left and 1 green one.

Probability of picking green is 1 out of 7 = 1/7

Probability of picking all 3 = 1/3 x 1/4 x 1/7 = 1/84

Answer:

Heather pay will be 290 for 40 hours of work.

Step-by-step explanation:

Given:

Amount he gets paid weekly =174

Number of hours of work =24

we need to find the amount of pay for 40 hours of work.

Also Given:

weekly pay is directly proportional to the number of hours.

Framing in equation form we get;

Amount Of Pay ∝ Number of hours of work

Hence Amount of Pay = k × Numbers of hours of work.

where k is constant.

Substituting the values we will find the value of k

[tex]174 = k \times 24\\\\k = \frac{174}{24} = 7.25[/tex]

Now using this we will find the amount of pay when hours of work is 40.

Amount Of Pay = [tex]7.25\times 40 = 290[/tex]

Hence Heather pay will be 290 for 40 hours of work.

Answer:

Step-by-step explanation:

Annual interest rate is 2.75%. Hence, the monthly interest rate is [tex]\frac{2.75}{12}[/tex]

The amount will be compounded [tex](20\times12) = 240[/tex] times.

Every month they deposits $500.

In the first month that deposited $500 will be compounded 240 times.

It will be [tex]500\times [1 + \frac{2.75}{1200} ]^{240}[/tex]

In the second month $500 will be deposited again, this time it will be compounded 239 times.

It will give [tex]500\times [1 + \frac{2.75}{1200} ]^{239}[/tex]

Hence, the total after 20 years will be [tex]500\times [1 + \frac{2.75}{1200} ]^{240} + 500\times [1 + \frac{2.75}{1200} ]^{239} + ........+ 500\times [1 + \frac{2.75}{1200} ]^{1} = 160110.6741[/tex]

The account will have approximately $159,744.59 after 20 years.

To calculate the future value of the savings account after 20 years with a 2.75% annual interest rate compounded monthly, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

Plugging in the values, we can calculate:

A = 500(1 + 0.0275/12)^(12*20)

A = 500(1.00229167)^(240)

A ≈ $159,744.59

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Answer:

Step-by-step explanation:

Mrs Cain coleslaw recipe calls for 1/3 cup of oil, 1/2 cup of vinegar, 1/4 cup of sugar. This means that the ratio of the number of cups of oil to vinegar to sugar is 1/3 : 1/2: 1/4

If she has 1 cup of vinegar, the ratio that would correspond to the given ratio would be determined by multiplying the given ratio by 2. It becomes, 2/3 : 2/2 : 2/4 = 2/3 : 1 : 1/2

Therefore, she would require 2/3 cups if oil and 1/2 cups of sugar to make 1 batch.

Answer:

The sprinkler must rotate by an angle of 107.48°.

Step-by-step explanation:

Given:

Area of strawberry patch( in shape of sector)  = 1500 square yards

Radius of circle = 40 yards

To find angle through which the sprinkler should rotate.

Solution.

In order to find the angle of rotation of sprinkler, we will apply the area of sector formula.

[tex]Area\ of\ sector\ = \frac{\theta}{360}\times \pi r^2[/tex]

where [tex]\theta[/tex] is the angle of the sector formed and [tex]r[/tex] is radius of the circle.

Thus, we can plugin the given values to find [tex]\theta[/tex] which would be the angle of rotation.

[tex]1500 = \frac{\theta}{360}\times \pi (40)^2[/tex]

Taking [tex]\pi=3.14[/tex]

[tex]1500 = \frac{\theta}{360}\times \ (3.14) (40)^2[/tex]

[tex]1500 = \frac{\theta}{360}\times 5024[/tex]

Dividing both sides by 5024.

[tex]\frac{1500}{5024} = \frac{\theta}{360}\times 5024\div 5024[/tex]

Multiplying both sides by 360.

[tex]\frac{1500\times 360}{5024} =\frac{\theta}{360}\times 360[/tex]

[tex]107.48=\theta[/tex]

∴ [tex]\theta= 107.48\°[/tex]

Angle of rotation of sprinkler = 107.48°

The strawberry farmer needs the sprinkler to rotate approximately 107.46 degrees to cover the 1500 square yard sector of the circle with a 40-yard radius.

To determine the angle through which the sprinkler should rotate, we first need to find the area of the sector of the circle. The formula for the area of a sector is:

A = 0.5 × r² × θ

where A is the area, r is the radius, and θ is the angle in radians. Here, we know the area A is 1500 square yards and the radius r is 40 yards. Rearranging the formula to solve for θ gives:

θ = (2 × A) / r²

Substituting the given values:

θ = (2 × 1500) / 40²

θ = (3000) / 1600

θ = 1.875 radians

To convert this angle in radians to degrees, we use the conversion factor 180/π:

θ = 1.875 × (180 / π) ≈ 107.46 degrees

Therefore, the sprinkler should rotate through an angle of approximately 107.46 degrees.

Good evening ,

Answer:

2x² - 18x + 36

Step-by-step explanation:

2(x-3)(x-6) = 2x²-18x+36.

:)

Answer:

[tex]cos(x)=\frac{\sqrt{5}}{5}[/tex]

Step-by-step explanation:

we have that

[tex]tan(x)=-2[/tex]

The angle x is in quadrant IV

That means ---> The value of cos(x) is positive and the value of sin(x) is negative

Remember that

[tex]cos^2(x)+sin^2(x)=1[/tex] ----> equation A

[tex]tan(x)=\frac{sin(x)}{cos(x)}[/tex]

so

[tex]-2=\frac{sin(x)}{cos(x)}[/tex]

[tex]sin(x)=-2cos(x)[/tex] ----> equation B

substitute equation B in equation A

[tex]cos^2(x)+(-2cos(x))^2=1[/tex]

solve for cos(x)

[tex]cos^2(x)+4cos^2(x)=1[/tex]

[tex]5cos^2(x)=1[/tex]

[tex]cos^2(x)=\frac{1}{5}[/tex]

square root both sides

[tex]cos(x)=\pm\frac{1}{\sqrt{5}}[/tex]

but remember that the value of cos(x) is positive (IV quadrant)

[tex]cos(x)=\frac{1}{\sqrt{5}}[/tex]

simplify

[tex]cos(x)=\frac{\sqrt{5}}{5}[/tex]

Answer:

0.2288

Step-by-step explanation:

This is straightforward.

What we need to do is to divide the number of defective semiconductor chips over the total number of semiconductor chips.

Initially, the total number of semi conductor chips is 119 and the total number of defective semiconductor chips is 28.

After the first selections , we can infer that the total number of semiconductor chips is 118 while the number of defective ones is 27.

Hence on the second drawing, the probability that he will

Select a defective one is 27/118

Answer:

11 positive integers can be expressed.

Step-by-step explanation:

Consider the provided information.

The number of possible prime numbers are 5,7,11,and 13.

There are 4 possible prime numbers.

How many positive integers can be expressed as a product of two or more of the prime numbers, that means there can be product of two numbers, three number or four numbers.

The formula to calculate combinations is: [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

The number of ways are:

[tex]^4C_2+^4C_3+^4C_4=\frac{4!}{2!(4-2)!}+\frac{4!}{3!(4-3)!}+\frac{4!}{4!}[/tex]

[tex]^4C_2+^4C_3+^4C_4=\frac{4!}{2!2!}+\frac{4!}{3!}+1[/tex]

[tex]^4C_2+^4C_3+^4C_4=6+4+1[/tex]

[tex]^4C_2+^4C_3+^4C_4=11[/tex]

Hence, 11 positive integers can be expressed.

the answer of this question is 603.2yd2

Answer:Area of the shaded region is 603.2 yards^2

Step-by-step explanation:

The diagram contains two circles. The smaller circle has a radius of 8 yards.

The bigger circle has a radius of 16 yards.

The area of a circle is expressed as

Area of circle = πr^2

Where

r = radius of circle

π is a constant whose value is 3.142

The area of the smaller circle would be

3.142 × 8^2 = 201.088 yards^2

The area of the bigger circle would be

3.14 × 16^2 = 804.352 yards^2

Area of the shaded region would be area of the bigger circle - area of the smaller circle. It becomes

804.352 - 201.088 = 603.2 yards^2

Answer:

The remainder is 64

Step-by-step explanation:

Consider the function [tex]f(x)=x^3+3x^2-51x+91[/tex]

Use synthetic division to divide by x+9

x+9=0, x=-9

divide the given f(x) by -9 using synthetic division

-9               1           3       -51       91

                  0          -9     54        -27              

         --------------------------------------------------

                  1           -6      3         64

The remainder is 64

Answer:

[tex]I_2=9[/tex]

Step-by-step explanation:

We have been told that the ratios are inversely proportional in our given problem. We are asked to find the missing value.

We know that two inversely proportional quantities are in form [tex]y=\frac{k}{x}[/tex], where, y is inversely proportional to x and k is the constant of proportionality.

Let us find constant of proportionality using [tex]R_1 = 6[/tex] and [tex]I_1 = 12[/tex] in above equation.

[tex]6=\frac{k}{12}[/tex]

[tex]6*12=\frac{k}{12}*12[/tex]

[tex]72=k[/tex]

Now, we will use [tex]72=k[/tex] and [tex]R_2 = 8[/tex] in our equation to find [tex]I_2[/tex] as:

[tex]8=\frac{72}{I_2}[/tex]

[tex]I_2=\frac{72}{8}[/tex]

[tex]I_2=9[/tex]

Therefore, the value of [tex]I_2[/tex] is 9.

The question is about the mathematical concept of inverse proportionality. The missing value of [tex]I_2[/tex] is found by applying the property of inversely proportional quantities, yielding the result [tex]I_2 = 9[/tex].

In an inverse proportionality relationship, the product of the values in one set is equal to the product of the corresponding values in the other set. This can be represented as:

[tex]\[R_1 \cdot I_1 = R_2 \cdot I_2\][/tex]

Given that [tex]\(R_1 = 6\)[/tex], [tex]\(R_2 = 8\)[/tex], and [tex]\(I_1 = 12\)[/tex], you can solve for [tex]\(I_2\)[/tex]:

[tex]\[6 \cdot 12 = 8 \cdot I_2\][/tex]

Now, simplify the equation:

[tex]\[72 = 8 \cdot I_2\][/tex]

To isolate [tex]\(I_2\)[/tex], divide both sides by 8:

[tex]\[I_2 = \frac{72}{8}\][/tex]

[tex]\[I_2 = 9\][/tex]

So, the value of [tex]\(I_2\)[/tex] is 9. In this inverse proportionality relationship, when [tex]R_1[/tex] is 6, [tex]\(R_2\)[/tex] is 8, and [tex]\(I_1\)[/tex] is 12, [tex]\(I_2\)[/tex] is 9. This means that as one variable increases, the other variable decreases in such a way that their product remains constant.

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Answer:

The answer is k=77

Step-by-step explanation:

It is given that n is an integer and greater than 2. Let us simplify k = (n + 2)(n – 2) to k=n^2-4.

In (1) it is stated that k is the product of two primes and in (2) it is stated that k<100. To solve the question we need to consider both cases (1) and (2).

Let say n=9 then k=n^2-4 becomes k=81-4=77. Well, 77 is the product of two prime numbers 7 and 11 (77=7*11). The answer is k=77

Answer:

  E.  1/2

Step-by-step explanation:

Divide by 2, then make use of the Pythagorean identity for sine and cosine.

  sin(x)^2 +cos(x)^2 = 2a

  1 = 2a . . . . . . . sin²+cos²=1

  1/2 = a

Answer:

Option E) is correct.

[tex]a=\frac{1}{2}[/tex]

Step-by-step explanation:

Given trignometric equation is [tex]2sin^{2}x + 2cos^{2}x = 4a[/tex]

To find the value of "a" from the given equation:

[tex]2sin^{2}x + 2cos^{2}x = 4a[/tex]

Taking common number "2" outside the equation of left hand side

[tex]2(sin^{2}x +cos^{2}x) = 4a[/tex]

[tex]sin^{2}x +cos^{2}x =\frac{4a}{2}[/tex]

[tex]sin^{2}x+cos^{2}x =2a[/tex]

( We know the trignometric formula  [tex]sin^{2}\theta +cos^{2}\theta=1[/tex] here

[tex]\theta=x[/tex]  )

Therefore  [tex](1) =2a[/tex]

[tex]\frac{1}{2} =a[/tex]

It can be written as

[tex]a=\frac{1}{2}[/tex]

Therefore  [tex]a=\frac{1}{2}[/tex]

Option E) is correct.

Answer: the number of new light bulbs that they put in is 36

Step-by-step explanation:

The Marino's put new light bulbs in all the fixtures the new house.

Each fixture used 2 light bulbs and each room had 3 fixtures. This means that the number of bulbs in each room would be 3×2 = 6 bulbs.

The new house had 6 rooms. This means that the total number of bulbs in 6 rooms would be

6 × 6 = 36 bulbs

Answer:

Remainder would be [tex]5x^2+21[/tex]

Step-by-step explanation:

Given,

Dividend = [tex]x^5+x^4+x^3+x^2+x[/tex]

Divisor = [tex]x^3-4x[/tex]

Using long division ( shown below ),

We get,

[tex]\frac{x^5+x^4+x^3+x^2+x}{x^3-4x}=x^2+x+5+\frac{5x^2+21}{x^3-4x}[/tex]

Therefore,

Remainder would be [tex]5x^2+21[/tex]

Answer:

[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:

[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]

Step-by-step explanation:

From the info given by the problem we need an integer defined as the smallest positive integer that is a multiple of 75 and have 75 positive integral divisors, and we are assuming that 1 is one possible divisor.

Th first step is find the prime factorization for the number 75 and we see that

[tex]75=3 5^2[/tex]

And we know that 3 =2+1 and 5=3+2 and if we replace we got:

[tex] 75 = (2+1)(4+1)^2 = (2+1)(4+1)(4+1)[/tex]

And in order to find 75 integral divisors we need to satisify this condition:

[tex]n= a^{r_1 -1}_1 a^{r_2 -1}_2 *......[/tex] such that [tex]a_1 *a_2*....=75[/tex]

For this case we have two prime factors important 3 and 5. And if we want to minimize n we can use a prime factor like 2. The least common denominator between 2 and 4 is LCM(2,4) =4. So then the need to have the prime factors 2 and 3 elevated at 4 in order to satisfy the condition required, and since 5 is the highest value we need to put the same exponent.

And then the value for n would be given by:

[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:

[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]

The smallest positive integer that is a multiple of 75 is 32400 and

integral divisors are 432.

Positive integers are the whole number that is greater than zero and do not include decimal or fraction values.

Find the smallest positive integer that is a multiple of 75 that has exactly 75 positive integral divisors.

We know that

[tex]75 = 3*5^{2}[/tex]

So the value of n which has 75 divisors then the multiplication of power of prime factor should be 75. The formula is given by

[tex]n = 2^{x-1} *3^{y-1} *5^{z-1}[/tex]

then the multiplication of x, y, and z must be 75.

x, y, and z are 5, 5, and 3 will be the values.

[tex]n = 2^{4} *3^{4} *5^{2} = 32400[/tex]

And it is divisible by 75 also.

[tex]\dfrac{n}{3*5^{2} } = \dfrac{2^{4} *3^{4} *5^{2}}{3*5^{2} } = 432[/tex]

Thus,  the smallest positive integer that is a multiple of 75 is 32400 and integral divisors are 432.

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Answer:

The answer to your question is letter A

Step-by-step explanation:

Process

1.- Find two points of the line

 A ( -2, 0)

 B ( 0, 3)

2.- Find the slope

     [tex]m = \frac{y2 - y1}{x2 - x1}[/tex]

            [tex]m = \frac{3 - 0}{0 + 2}[/tex]

                   [tex]m = \frac{3}{2}[/tex]

3.- Find the equation of the line

            y - y1 = m(x - x1)

            y - 0 = [tex]\frac{3}{2} (x + 2)[/tex]

            y = [tex]\frac{3}{2} (x + 2)[/tex]

Simplify

            y = [tex]\frac{3}{2} x + 3[/tex]

Y is a bisector of the two sides, so would be half the length of the base.

Y = 36 / 2

y = 18

Answer:

[tex]h=\frac{SA-2s^2}{4s}[/tex]

Step-by-step explanation:

we know that'

The formula to calculate the surface area of a square prism is

[tex]SA=2s^2+4sh[/tex]

where

s is the length of the side of the square base

h is the height of the prism

Solve for h

That means ----> Isolate the variable h

so

subtract 2s^2 both sides

[tex]SA-2s^2=4sh[/tex]

Divide by 4s both sides

[tex]\frac{SA-2s^2}{4s}=h[/tex]

Rewrite

[tex]h=\frac{SA-2s^2}{4s}[/tex]

To find the height of a square prism when given the surface area and base side length, use the formula B. H = (SA - 2s^2) / (4s).

The question asks for a formula that can be used to find the height (h) of a square prism given the surface area (SA) and the side length of the base (s). The surface area of a square prism is calculated using the formula SA = 2s2 + 4sh. To solve for h, we need to re-arrange this equation:

Therefore, the correct formula to find h is B. H = (SA – 2s2) / 4s.

Answer:

(1) Insufficient data

(2) Sufficient data

Step-by-step explanation:

We need to check whether the given data is sufficient or not to prove that a rectangular region has perimeter P inches and area A square inches, is the region square.

Assume that the given conditions are

1. [tex]P=\frac{4}{3}A[/tex]

2. [tex]P=4\sqrt{A}[/tex]

Area of square is

[tex]A=a^2[/tex]

Taking square root on both sides.

[tex]\sqrt{A}=a[/tex]             .... (1)

where a is side length.

Perimeter of square is

[tex]P=4a[/tex]            ... (2)

From (1) and (2) we get

[tex]P=4\sqrt{A}[/tex]

It means second condition is sufficient to prove that the rectangle reason is square, because it is true for all.

For a=3,

A=9 and P=12

[tex]12=\frac{4}{3}(9)[/tex]

[tex]12=12[/tex]

LHS=RHS

For a=6,

A=36 and P=24

[tex]24=48[/tex]   (False statement)

The first condition is insufficient because it may or may not be true.

The store received 1725 pencils and 1350 pens as of the given conditions.

Given that,
A store receives a shipment of pens and pencils. Each box contains 75 items There are 23 boxes of pencils and 18 boxes of pens. How many pens and pencils the store receive is to be determined.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

Here,
Each box contains 75 items,
The number of item in 23 pencil boxes = 75 × 23 = 1725 items,
The number of items in 18 pencil boxes = 18 × 75 = 1350 items.

Thus, the store received 1725 pencils and 1350 pens as of the given conditions.

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The cost of the bike for the given condition will be $230.

Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.

In other words, an equation is a set of variables that are constrained through a situation or case.

Suppose the cost of one bike is x while accessories for one bike is y.

As per the given,

x + y = 276

The cost of bike is 5 times more then accessories,

x = 5y

5y + y = 276

6y = 276

y = 46

Bike cost = 5(46) = $230

Hence "The bike will cost $230 in the specified condition".

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Answer:

P(X1=1, X2=1) = 1/15

P(X1=1, X2=0) = 7/30

P(X1=0, X2=1) = 7/30

P(X1=0, X2=0) = 7/15

Step-by-step explanation:

Let Xi = 1 if the i-th white ball is selected. In this question the 3 white balls are marked 1,2 and 3.

We need to know the possible combination between X1 and X2 i.e. for the white ball 1 and 2 being chosen in the event.

We also need to note that the event is dependent which that after a ball is being chosen, it will not be put back hence affecting the probability of picking the next ball.

Consider all the possible combination between X1 and X2

a) both being chosen P(X1=1, X2=1)

= (3/10) x (2/9) = 1/15

Note that the first probability is the probability before any ball is being picked. The chances for ball white to be pick is 3/10 (3 white ball from the total 10 balls).

After 1 white ball being selected, that ball is not again out back into the urn making white ball 2 and total ball 9. Hence probability of picking another white ball is 2/9

b) only X1 chosen P(X1=1, X2=0)

= (3/10) x (7/9) = 7/30

After the white ball was picked, the probability of white not being pick again is the same as red being picked. Since there is still 7red balls and a total of 9 balls, the probability is 7/9

c) only X2 chosen P(X1=0, X2=1)

= (7/10) x (3/9) = 7/30

The white is not being picked first, making the probability of picking red is 7/10. Then the probability of white being picked is 3/9

d) both not chosen

P(X1=0, X2=0)

= (7/10) x (6/9) = 7/15

In other word only red being chosen. So the first probability is 7 red out of 10 balls (7/10), and the next red ball being picked next is 6/9

Answer:

[tex]a_{n+1}=0.2a_n[/tex] for all n>0, [tex]a_1=16[/tex]

Step-by-step explanation:

Let [tex]\{a_n\}=\{16,3.2,0.64,0.128,\cdots \}[/tex] be the sequence described.

A geometric sequence has the following property: there exists some r (the ratio of the sequence) such that [tex]\frac{a_{n+1}}{a_n}=r[/tex] forr all n>0.

To find r, note that

[tex]\frac{3.2}{16}=\frac{32}{10(16)}=\frac{2}{10}=\frac{1}{5}=0.2[/tex]

Similarly

[tex]\frac{0.64}{3.2}=\frac{64}{10(32)}=\frac{1}{5}=0.2[/tex]

[tex]\frac{0.128}{0.64}=\frac{1}{5}=0.2[/tex]

Thus [tex]a_{n+1}=r a_n=\frac{a_n}{5}=0.2a_n[/tex] for all n>0, and [tex]a_1=16[/tex]

first term = 16

average ratio = 0.2

Answer:

[tex]P(X\geq 5)=1-P(X< 5)=1-[0.000649+0.00703+0.0343+0.0991+0.1878]=0.6712[/tex]

Step-by-step explanation:

1) Previous concepts  

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".  

2) Solution to the problem  

Let X the random variable of interest, on this case we now that:  

[tex]X \sim Binom(n=10, p=0.52)[/tex]  

The probability mass function for the Binomial distribution is given as:  

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]  

Where (nCx) means combinatory and it's given by this formula:  

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]  

What is the probability that at least 5 of the students in your study group of 10 have studied in the last week?

[tex]P(X\geq 5)=1-P(X< 5)=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)][/tex]

[tex]P(X=0)=(10C0)(0.52)^0 (1-0.52)^{10-0}=0.000649[/tex]  

[tex]P(X=1)=(10C1)(0.52)^1 (1-0.52)^{10-1}=0.00703[/tex]  

[tex]P(X=2)=(10C2)(0.52)^2 (1-0.52)^{10-2}=0.0343[/tex]  

[tex]P(X=3)=(10C3)(0.52)^3 (1-0.52)^{10-3}=0.0991[/tex]  

[tex]P(X=4)=(10C4)(0.52)^4 (1-0.52)^{10-4}=0.1878[/tex]  

[tex]P(X\geq 5)=1-P(X< 5)=1-[0.000649+0.00703+0.0343+0.0991+0.1878]=0.6712[/tex]

To find the probability that at least 5 of the students in a study group of 10 have studied in the last week, use the binomial probability formula and calculate the respective probabilities for each case. Add these probabilities together to get the final probability.

To calculate the probability that at least 5 of the students in your study group of 10 have studied in the last week, we can use the binomial probability formula. Let's denote the probability that a randomly selected teenager studied at least once during the week as p = 0.52. We want to find P(X >= 5) where X represents the number of teenagers in the study group who studied.

Using the binomial probability formula, P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10). We can calculate each of these individual probabilities using the formula: [tex]P(X = k) = C(n, k) * p^k * (1-p)^(^n^-^k^),[/tex] where C(n, k) is the combination of n items taken k at a time.

Once we have calculated each of these probabilities, we can add them together to find the final probability.

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Answer:

D

Step-by-step explanation:

Finding y int by substituting the points given (say y int is x)

(-9) = 4(-4) + x

(-9) = -16 + x

x = 7

With the y int, you can now write the equation

y = -4x + 7

The equation of the line will be y = –4x + 7 having a slope of –4 that passes through the point (4, –9). Option D is correct.

To find the equation of a line that has a slope of –4, and includes the point (4, –9), we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Plugging in our values, we get y - (-9) = -4(x - 4). Simplifying this, we have:

Subtract 9 from both sides to solve for y:

Thus, the correct equation is y = –4x + 7.

Hence, D. is the correct option.